🧠:<think>Okay, so I have this problem about an entrepreneur who runs craft beer bars, and they've got some data on customer visits. The problem is split into two parts: one about seasonality and another about market trends. I need to figure out the function for the number of visits considering seasonality and then calculate the expected average visits after two years considering market trends. Let me try to break this down step by step.Starting with the first part: the seasonality factor. It says that the number of visits V(t) can be modeled by a sinusoidal function because of seasonal trends. The peak season results in a 30% increase compared to the lowest season, and the cycle repeats every 6 months. The average number of visits per month is 1000, and we need to find V(t) where t is the time in months, starting from the lowest season.Hmm, okay. So sinusoidal functions are like sine or cosine waves. Since they mention starting from the lowest season, maybe a cosine function would be appropriate because cosine starts at its maximum, but if we shift it, we can have it start at the minimum. Alternatively, we can use a sine function with a phase shift. Let me think.First, let's recall the general form of a sinusoidal function:V(t) = A * sin(Bt + C) + DorV(t) = A * cos(Bt + C) + DWhere A is the amplitude, B affects the period, C is the phase shift, and D is the vertical shift.But since we're starting at the lowest season, maybe a cosine function with a phase shift of π would work because cos(t + π) is equal to -cos(t), which starts at -1, the minimum. Alternatively, we can use a sine function with a phase shift of -π/2, which would also start at the minimum.But maybe it's simpler to use a cosine function and just adjust the vertical shift accordingly. Let's see.The average number of visits is 1000. In sinusoidal functions, the vertical shift D is the average value. So D = 1000.Now, the amplitude A is the maximum deviation from the average. The peak season is a 30% increase over the lowest season. So if the lowest season is the minimum value, and the peak is 30% higher than that.Let me denote the minimum number of visits as V_min and the maximum as V_max.Given that V_max = V_min + 0.3 * V_min = 1.3 * V_min.But the average is 1000, which is the midpoint between V_min and V_max. So:(V_min + V_max)/2 = 1000Substituting V_max = 1.3 * V_min:(V_min + 1.3 * V_min)/2 = 1000(2.3 * V_min)/2 = 10001.15 * V_min = 1000So V_min = 1000 / 1.15 ≈ 869.565Therefore, V_max = 1.3 * V_min ≈ 1.3 * 869.565 ≈ 1130.435So the amplitude A is the difference between the average and the maximum (or minimum). Since the average is 1000, the amplitude is 1000 - V_min ≈ 1000 - 869.565 ≈ 130.435.Alternatively, since the function oscillates between V_min and V_max, the amplitude is half the difference between V_max and V_min.A = (V_max - V_min)/2 ≈ (1130.435 - 869.565)/2 ≈ 260.87 / 2 ≈ 130.435. Yep, same result.So A ≈ 130.435.Now, the period of the function is 6 months because the cycle repeats every 6 months. The general period of a sinusoidal function is 2π / |B|. So if the period is 6, then:2π / B = 6 => B = 2π / 6 = π / 3 ≈ 1.0472.So B is π/3.Now, since we're starting from the lowest season at t=0, we need to make sure that the function is at its minimum when t=0. Let's consider using a cosine function because cosine starts at its maximum, but if we invert it, it would start at its minimum.So let's write the function as:V(t) = A * cos(Bt + C) + DWe need V(0) = V_min. So plugging t=0:V(0) = A * cos(C) + D = V_minWe know that D = 1000, A ≈ 130.435, V_min ≈ 869.565.So:130.435 * cos(C) + 1000 = 869.565130.435 * cos(C) = 869.565 - 1000 = -130.435So cos(C) = -130.435 / 130.435 = -1Therefore, C = π.So the function becomes:V(t) = 130.435 * cos( (π/3) * t + π ) + 1000Alternatively, since cos(θ + π) = -cos(θ), we can write:V(t) = -130.435 * cos( (π/3) * t ) + 1000But let me check if this makes sense. At t=0:V(0) = -130.435 * cos(0) + 1000 = -130.435 * 1 + 1000 ≈ 869.565, which is correct.At t=3 (half period), cos( (π/3)*3 ) = cos(π) = -1, so V(3) = -130.435*(-1) + 1000 ≈ 130.435 + 1000 ≈ 1130.435, which is the peak. That seems correct.Alternatively, if I use a sine function, I might need a phase shift. Let's see:V(t) = A * sin(Bt + C) + DWe want V(0) = V_min, so:130.435 * sin(C) + 1000 = 869.565sin(C) = (869.565 - 1000) / 130.435 ≈ (-130.435)/130.435 = -1So C = -π/2, because sin(-π/2) = -1.So the function would be:V(t) = 130.435 * sin( (π/3)*t - π/2 ) + 1000This also works because at t=0, sin(-π/2) = -1, so V(0) = -130.435 + 1000 ≈ 869.565.But the cosine function with phase shift π is simpler, I think. So I'll stick with that.So, V(t) = 130.435 * cos( (π/3)*t + π ) + 1000Alternatively, as I noted before, this is equivalent to:V(t) = -130.435 * cos( (π/3)*t ) + 1000But maybe it's better to write it in terms of a cosine function without the negative sign, so perhaps:V(t) = 130.435 * cos( (π/3)*t + π ) + 1000Either way is correct, but perhaps the first form is more straightforward.Wait, actually, let me check the amplitude again. The amplitude is 130.435, which is approximately 130.44. But maybe we can express this exactly.Since V_min = 1000 / 1.15, which is 1000 * (20/23) ≈ 869.565, and V_max = 1000 * (23/20) ≈ 1150.Wait, hold on, 1.3 times V_min is V_max. So V_max = 1.3 * V_min.But V_min = 1000 - A, and V_max = 1000 + A.So 1000 + A = 1.3 * (1000 - A)1000 + A = 1300 - 1.3AA + 1.3A = 1300 - 10002.3A = 300A = 300 / 2.3 ≈ 130.4348So exactly, A = 300 / 2.3 = 130.4347826...So, to keep it exact, maybe we can write A as 300/2.3, but that's a repeating decimal. Alternatively, we can write it as 3000/23, which is exact.So 3000/23 ≈ 130.4348.So, perhaps, to keep it exact, we can write A as 3000/23.So, V(t) = (3000/23) * cos( (π/3)*t + π ) + 1000Alternatively, since cos(θ + π) = -cos(θ), we can write:V(t) = - (3000/23) * cos( (π/3)*t ) + 1000But let me check if this is necessary. The problem says to find the function V(t). It doesn't specify whether to use sine or cosine, so either is acceptable as long as it's correct.So, in summary, the function is a cosine function with amplitude 3000/23, period 6 months, shifted vertically by 1000, and phase shifted by π to start at the minimum.So, V(t) = (3000/23) * cos( (π/3)*t + π ) + 1000Alternatively, V(t) = - (3000/23) * cos( (π/3)*t ) + 1000Both are correct.Now, moving on to the second part: market trend analysis. The entrepreneur tracks market trends using a linear model. The average growth rate of customer visits due to market trends alone has been 5% per quarter. Assuming this growth rate continues, ignoring seasonality, calculate the expected average number of visits per month at the end of 2 years, starting from an initial average of 1000 visits per month.Okay, so this is a linear growth model? Wait, but 5% per quarter is a growth rate, which is typically compounded. So is it linear or exponential?Wait, the problem says "linear model," but a 5% growth rate per quarter is usually exponential growth. Hmm, maybe I need to clarify.Wait, let me read again: "the average growth rate of customer visits due to market trends alone has been 5% per quarter." So, if it's a linear model, that would mean the increase is 5% per quarter in an additive way, but that doesn't make much sense because percentages are multiplicative.Wait, maybe it's a linear growth in terms of the number of visits, not percentage. But 5% per quarter is a percentage, so perhaps it's exponential growth.Wait, the problem says "linear model," but the growth rate is given as a percentage. That seems contradictory. Maybe it's a linear growth in terms of the logarithm, but that's not standard.Wait, perhaps it's a linear increase in the number of visits, not percentage. So, 5% per quarter of the initial amount? That would be linear growth.Wait, let's think. If it's linear, then the growth is constant per quarter. So, if the initial average is 1000, and the growth rate is 5% per quarter, then each quarter, the visits increase by 5% of 1000, which is 50 visits. So, each quarter, it's 50 more visits. So, over 2 years, which is 8 quarters, the total increase would be 8*50 = 400, so the average would be 1000 + 400 = 1400.But that seems like a linear growth model where the increase is constant each quarter.Alternatively, if it's exponential growth, then each quarter, the visits are multiplied by 1.05. So, after 8 quarters, it would be 1000*(1.05)^8.But the problem says "linear model," so maybe it's the additive model, not multiplicative. So, 5% of 1000 is 50, so each quarter, add 50.Therefore, after 2 years (8 quarters), the average would be 1000 + 8*50 = 1400.But let me double-check. The term "linear model" in this context probably refers to linear growth, meaning constant increase per period, rather than exponential growth. So, if the growth rate is 5% per quarter, that would mean 5% of the initial amount each quarter, which is 50 per quarter.So, over 8 quarters, that's 8*50 = 400, so total is 1400.Alternatively, if it's 5% per quarter on the current amount, that would be exponential. But the problem says "linear model," so I think it's the former.Wait, but let's think about what a linear model would imply. A linear model would have the form V(t) = V0 + rt, where r is the rate. So, if the rate is 5% per quarter, that would mean r = 0.05*V0 per quarter, but that would make it a linear function with a slope of 0.05*V0.But in this case, V0 is 1000, so r = 50 per quarter. So, yes, V(t) = 1000 + 50*t, where t is in quarters.Therefore, after 2 years, which is 8 quarters, V(8) = 1000 + 50*8 = 1400.Alternatively, if it's exponential, it would be V(t) = 1000*(1.05)^t, where t is quarters. So, after 8 quarters, V(8) = 1000*(1.05)^8 ≈ 1000*1.477455 ≈ 1477.46.But since the problem specifies a linear model, I think the first interpretation is correct, so the answer is 1400.Wait, but let me check the wording again: "the average growth rate of customer visits due to market trends alone has been 5% per quarter." So, if it's an average growth rate, that could imply that it's compounded, i.e., exponential, because growth rates are typically compounded. But the problem says "linear model," which is confusing.Hmm, this is a bit ambiguous. Let me think about what a linear model would mean in this context. A linear model would mean that the increase is constant each period, not proportional to the current value. So, if it's 5% per quarter, that would mean 5% of the initial value each quarter, which is 50 per quarter, leading to a linear increase.Alternatively, if it's 5% per quarter on the current value, that's exponential growth, which is a multiplicative model, not linear.Given that the problem says "linear model," I think it's the former: a constant increase of 5% of the initial amount each quarter.So, with that, the expected average number of visits per month at the end of 2 years would be 1400.But wait, the initial average is 1000 visits per month. If the growth is 5% per quarter, and it's linear, then each quarter, the monthly average increases by 5% of 1000, which is 50. So, each quarter, the monthly average goes up by 50.But wait, hold on. If it's 5% per quarter, and it's linear, does that mean 5% per quarter of the initial value, or 5% per quarter of the current value?If it's linear, it's additive, so it's 5% of the initial value each quarter. So, 50 per quarter.But if it's exponential, it's multiplicative, so each quarter, it's 5% more than the previous quarter.But the problem says "linear model," so I think it's additive.Therefore, over 8 quarters, the increase is 8*50 = 400, so the average is 1400.Alternatively, if it's exponential, it's 1000*(1.05)^8 ≈ 1477.46.But since the problem specifies a linear model, I think it's 1400.Wait, but let me think again. In finance, a growth rate is typically exponential, but in some contexts, people might refer to linear growth when they mean constant absolute increases. So, perhaps, in this case, since it's a market trend analysis, and they mention a linear model, it's more likely to be additive.So, I think the answer is 1400.But just to be thorough, let me calculate both and see which one makes sense.If it's linear:V(t) = 1000 + 50*t, where t is quarters.After 2 years, t=8:V(8) = 1000 + 50*8 = 1400.If it's exponential:V(t) = 1000*(1.05)^t, t=8:V(8) ≈ 1000*1.477455 ≈ 1477.46.Given that the problem says "linear model," I think 1400 is the answer they're looking for.So, to recap:1. The seasonality function is V(t) = (3000/23) * cos( (π/3)*t + π ) + 1000, which simplifies to V(t) = - (3000/23) * cos( (π/3)*t ) + 1000.2. The expected average number of visits per month after 2 years, considering the linear market trend, is 1400.I think that's it. Let me just write the final answers clearly.</think>

🧠:<think>Alright, so I've got this problem about a cryptographic function, and I need to analyze its security. Let me try to break it down step by step.First, the function is defined as ( f(x) = x^e mod n ), where ( e ) and ( n ) are large coprime integers. ( x ) is an integer such that ( 0 leq x < n ). Also, ( n ) is given as the product of two distinct primes ( p ) and ( q ). So, ( n = pq ). The first part of the problem asks me to prove that if someone can find an integer ( y ) such that ( f(y) equiv f(x) mod n ) and ( y neq x ), then the security of this cryptographic scheme is compromised. Specifically, I need to show how finding such a ( y ) relates to factoring ( n ).Hmm, okay. So, ( f(y) equiv f(x) mod n ) implies that ( y^e equiv x^e mod n ). Which further implies that ( y^e - x^e ) is divisible by ( n ). Since ( n = pq ), this means that ( y^e equiv x^e mod p ) and ( y^e equiv x^e mod q ).Now, if ( y neq x ), then we have two different integers ( y ) and ( x ) such that their ( e )-th powers are congruent modulo ( p ) and modulo ( q ). So, ( y^e equiv x^e mod p ) and ( y^e equiv x^e mod q ).In modular arithmetic, if ( y^e equiv x^e mod p ), then ( (y x^{-1})^e equiv 1 mod p ). Let me denote ( z = y x^{-1} mod p ). Then, ( z^e equiv 1 mod p ). Similarly, ( z^e equiv 1 mod q ).So, ( z ) is an element of the multiplicative group modulo ( p ) and modulo ( q ). The multiplicative group modulo a prime ( p ) is cyclic of order ( p - 1 ). Therefore, the order of ( z ) modulo ( p ) divides both ( e ) and ( p - 1 ). Similarly, the order of ( z ) modulo ( q ) divides both ( e ) and ( q - 1 ).Since ( e ) is coprime to ( varphi(n) = (p - 1)(q - 1) ), it must be that ( e ) is coprime to both ( p - 1 ) and ( q - 1 ). Therefore, the only element with order dividing ( e ) in the multiplicative group modulo ( p ) is the identity element, which is 1. Similarly, modulo ( q ), the only such element is 1.Wait, but ( z^e equiv 1 mod p ) and ( z^e equiv 1 mod q ). If ( z ) is 1 modulo both ( p ) and ( q ), then ( z equiv 1 mod n ). But ( z = y x^{-1} mod n ), so ( y equiv x mod n ). But since ( y ) and ( x ) are both less than ( n ), this would imply ( y = x ). But we have ( y neq x ), so this leads to a contradiction.Hmm, so maybe my initial approach is missing something. Let me think again.If ( y^e equiv x^e mod n ), then ( y^e equiv x^e mod p ) and ( y^e equiv x^e mod q ). So, in each modulus, ( (y/x)^e equiv 1 mod p ) and ( (y/x)^e equiv 1 mod q ). Let me denote ( z = y/x mod p ) and ( z' = y/x mod q ). Then, ( z^e equiv 1 mod p ) and ( z'^e equiv 1 mod q ).Since ( e ) is coprime to ( p - 1 ) and ( q - 1 ), the only solution to ( z^e equiv 1 mod p ) is ( z equiv 1 mod p ), and similarly ( z' equiv 1 mod q ). Therefore, ( y equiv x mod p ) and ( y equiv x mod q ). By the Chinese Remainder Theorem, this implies ( y equiv x mod n ). But since ( y ) and ( x ) are both between 0 and ( n - 1 ), this would mean ( y = x ). But we have ( y neq x ), so this seems impossible.Wait, so if such a ( y ) exists, it must be that ( y equiv x mod n ), which would imply ( y = x ), contradicting ( y neq x ). Therefore, such a ( y ) cannot exist unless ( n ) is not the product of two distinct primes, or ( e ) is not coprime to ( varphi(n) ). But the problem states that ( e ) and ( n ) are coprime, which implies ( e ) is coprime to ( varphi(n) ) because ( n = pq ).Hmm, maybe I'm approaching this the wrong way. Perhaps the existence of such a ( y ) implies that ( e ) is not coprime to ( varphi(n) ), which would mean that the function ( f(x) ) is not injective, hence not invertible, which would compromise the security.Alternatively, maybe the existence of such a ( y ) allows us to find a non-trivial square root of 1 modulo ( n ), which can be used to factor ( n ).Wait, let's think about it. If ( y^e equiv x^e mod n ), then ( (y x^{-1})^e equiv 1 mod n ). So, ( z = y x^{-1} ) satisfies ( z^e equiv 1 mod n ). If ( z neq 1 mod n ), then ( z ) is a non-trivial ( e )-th root of unity modulo ( n ).But since ( e ) is coprime to ( varphi(n) ), the only solutions to ( z^e equiv 1 mod n ) should be ( z equiv 1 mod n ). Therefore, if such a ( z ) exists where ( z neq 1 mod n ), it would imply that ( e ) is not coprime to ( varphi(n) ), which contradicts the given condition.Wait, but the problem states that ( e ) and ( n ) are coprime. Since ( n = pq ), and ( e ) is coprime to ( n ), it doesn't necessarily mean ( e ) is coprime to ( varphi(n) ). Wait, actually, ( varphi(n) = (p - 1)(q - 1) ). So, ( e ) being coprime to ( n ) doesn't automatically make it coprime to ( varphi(n) ). For example, ( e ) could share a factor with ( p - 1 ) or ( q - 1 ).Ah, so if ( e ) is not coprime to ( varphi(n) ), then the function ( f(x) ) is not injective, meaning there exist distinct ( x ) and ( y ) such that ( f(x) = f(y) mod n ). Therefore, the existence of such a ( y ) would imply that ( e ) shares a common factor with ( varphi(n) ), which in turn implies that ( e ) is not coprime to ( varphi(n) ). But wait, the problem states that ( e ) and ( n ) are coprime, but doesn't explicitly state that ( e ) is coprime to ( varphi(n) ). So, perhaps the function is not necessarily injective, and finding such a ( y ) would mean that ( e ) is not coprime to ( varphi(n) ), which would allow us to factor ( n ).Alternatively, maybe the existence of such a ( y ) allows us to compute the factors ( p ) and ( q ). Let me think about that.Suppose we have ( y^e equiv x^e mod n ), but ( y neq x mod n ). Then, as before, ( (y x^{-1})^e equiv 1 mod n ). Let ( z = y x^{-1} mod n ). So, ( z^e equiv 1 mod n ). If ( z neq 1 mod n ), then ( z ) is a non-trivial ( e )-th root of unity modulo ( n ).Now, since ( n = pq ), we can consider ( z ) modulo ( p ) and modulo ( q ). Let ( z_p = z mod p ) and ( z_q = z mod q ). Then, ( z_p^e equiv 1 mod p ) and ( z_q^e equiv 1 mod q ).Since ( p ) and ( q ) are primes, the multiplicative groups modulo ( p ) and ( q ) have orders ( p - 1 ) and ( q - 1 ), respectively. Therefore, the order of ( z_p ) modulo ( p ) divides both ( e ) and ( p - 1 ). Similarly, the order of ( z_q ) modulo ( q ) divides both ( e ) and ( q - 1 ).If ( e ) is coprime to ( p - 1 ), then the only solution to ( z_p^e equiv 1 mod p ) is ( z_p equiv 1 mod p ). Similarly, if ( e ) is coprime to ( q - 1 ), then ( z_q equiv 1 mod q ). But if ( e ) shares a common factor with ( p - 1 ) or ( q - 1 ), then there could be non-trivial solutions.Wait, but the problem states that ( e ) and ( n ) are coprime, which means ( e ) is coprime to ( p ) and ( q ), but not necessarily to ( p - 1 ) or ( q - 1 ). So, if ( e ) shares a common factor with ( p - 1 ), say ( d ), then the multiplicative order of ( z_p ) modulo ( p ) divides ( d ), which is greater than 1. Similarly for ( q ).Therefore, if such a ( z ) exists, it implies that ( e ) shares a common factor with ( p - 1 ) or ( q - 1 ). This would allow us to compute the factors ( p ) and ( q ) by finding the order of ( z ) modulo ( n ), which can be used to find the factors.Alternatively, since ( z^e equiv 1 mod n ), we can use the fact that ( z ) is a non-trivial root of unity modulo ( n ) to find a non-trivial factor of ( n ). For example, we can compute ( gcd(z - 1, n) ), which would give us either ( p ) or ( q ).Wait, let's test this idea. Suppose ( z equiv 1 mod p ) but ( z notequiv 1 mod q ). Then, ( z - 1 equiv 0 mod p ), so ( gcd(z - 1, n) = p ). Similarly, if ( z equiv 1 mod q ) but not modulo ( p ), then ( gcd(z - 1, n) = q ). If ( z equiv 1 mod n ), then ( z = 1 ), which we've already considered.But in our case, ( z neq 1 mod n ), so either ( z equiv 1 mod p ) or ( z equiv 1 mod q ), but not both. Therefore, computing ( gcd(z - 1, n) ) would give us one of the prime factors of ( n ).Therefore, if such a ( y ) exists, we can find a non-trivial ( z ) such that ( z^e equiv 1 mod n ), and then use ( z ) to compute a factor of ( n ). This would compromise the security of the cryptographic scheme because factoring ( n ) allows us to compute the private key, which is necessary for decryption.So, to summarize, if someone can find a ( y neq x ) such that ( f(y) equiv f(x) mod n ), then they can find a non-trivial ( e )-th root of unity modulo ( n ), which can be used to factor ( n ) and thus break the security of the scheme.Now, moving on to the second part of the problem. It asks me to determine the conditions under which a pair of integers ( (e_1, e_2) ), both coprime to ( varphi(n) ), exist such that ( e_1 cdot e_2 equiv 1 mod varphi(n) ). I need to discuss the implications of this scenario on the invertibility of ( f(x) ) and how it might be used to reverse-engineer the one-way property.Okay, so ( e_1 ) and ( e_2 ) are both coprime to ( varphi(n) ), and their product is congruent to 1 modulo ( varphi(n) ). This means that ( e_2 ) is the multiplicative inverse of ( e_1 ) modulo ( varphi(n) ). Therefore, ( e_1 ) and ( e_2 ) form a pair of exponents that are inverses in the multiplicative group modulo ( varphi(n) ).In the context of the function ( f(x) = x^e mod n ), the invertibility of ( f ) depends on whether ( e ) is coprime to ( varphi(n) ). If ( e ) is coprime to ( varphi(n) ), then there exists an inverse exponent ( d ) such that ( e cdot d equiv 1 mod varphi(n) ), and thus ( f(f(x)^d) = x mod n ). This is the basis of RSA encryption, where ( e ) is the public exponent and ( d ) is the private exponent.So, if such a pair ( (e_1, e_2) ) exists, it means that ( e_1 ) has an inverse ( e_2 ) modulo ( varphi(n) ). Therefore, the function ( f(x) = x^{e_1} mod n ) is invertible, with the inverse function being ( f^{-1}(x) = x^{e_2} mod n ).The implication is that if both ( e_1 ) and ( e_2 ) are known, then the function ( f ) can be reversed, effectively making it not a one-way function. However, in a secure cryptographic system, the private exponent ( e_2 ) is kept secret, and only the public exponent ( e_1 ) is known. Therefore, the existence of such a pair doesn't inherently compromise security unless the private exponent is revealed.But if an attacker can find such a pair ( (e_1, e_2) ) without knowing the factors of ( n ), it would mean that they can invert the function ( f ), thereby breaking the one-way property. However, finding such a pair requires knowing ( varphi(n) ), which in turn requires factoring ( n ). Therefore, as long as factoring ( n ) is computationally infeasible, the function remains secure.Wait, but the problem states that both ( e_1 ) and ( e_2 ) are coprime to ( varphi(n) ). So, if ( e_1 ) is given, ( e_2 ) can be computed as the inverse modulo ( varphi(n) ). But to compute ( e_2 ), one needs to know ( varphi(n) ), which requires knowing the factors ( p ) and ( q ). Therefore, unless an attacker can factor ( n ), they cannot compute ( e_2 ), even if they know ( e_1 ).Therefore, the existence of such a pair ( (e_1, e_2) ) doesn't directly compromise the security unless the attacker can compute ( e_2 ) from ( e_1 ), which is equivalent to factoring ( n ). So, the invertibility of ( f(x) ) is tied to the knowledge of ( varphi(n) ), which is tied to the ability to factor ( n ).In summary, the function ( f(x) ) is invertible if and only if the exponent ( e ) is coprime to ( varphi(n) ), and the inverse function exists with exponent ( d ) such that ( e cdot d equiv 1 mod varphi(n) ). However, without knowing ( varphi(n) ), which requires factoring ( n ), the function remains one-way, maintaining its security.So, to answer the second part, the conditions for such a pair ( (e_1, e_2) ) to exist are that both ( e_1 ) and ( e_2 ) must be coprime to ( varphi(n) ), and ( e_1 ) must have an inverse modulo ( varphi(n) ), which is ( e_2 ). The implication is that if an attacker can find ( e_2 ) given ( e_1 ), they can invert the function, but this requires knowing ( varphi(n) ), which is equivalent to factoring ( n ). Therefore, as long as factoring ( n ) is hard, the function remains secure.Wait, but the problem says "both coprime to ( varphi(n) )", so ( e_1 ) and ( e_2 ) must each individually be coprime to ( varphi(n) ). Since ( e_2 ) is the inverse of ( e_1 ), and inverses modulo ( varphi(n) ) exist only if ( e_1 ) is coprime to ( varphi(n) ), so ( e_2 ) will automatically be coprime as well because it's the inverse.Therefore, the condition is simply that ( e_1 ) is coprime to ( varphi(n) ), which ensures the existence of ( e_2 ). The implication is that the function ( f(x) ) is invertible, but the security relies on the difficulty of finding ( e_2 ) without knowing ( varphi(n) ), which is tied to the difficulty of factoring ( n ).So, putting it all together, the existence of such a pair ( (e_1, e_2) ) means that the function is invertible, but the security of the scheme depends on the computational difficulty of factoring ( n ), which is assumed to be hard for large ( n ).I think I've covered both parts. Let me just recap:1. Finding a ( y neq x ) such that ( f(y) equiv f(x) mod n ) implies the existence of a non-trivial ( e )-th root of unity modulo ( n ), which can be used to factor ( n ), thus compromising security.2. The existence of a pair ( (e_1, e_2) ) with ( e_1 cdot e_2 equiv 1 mod varphi(n) ) implies that the function is invertible, but the security remains intact as long as factoring ( n ) is computationally infeasible.</think>

🧠:<think>Okay, so I have this problem about a booktuber analyzing the success of film adaptation videos based on views. There are two parts to the problem. Let me try to tackle them one by one.Starting with part 1: I need to calculate the probability that a randomly selected video has between 4900 and 5100 views inclusive. The number of views, ( V_i ), follows a Poisson distribution with a mean ( lambda = 5000 ). The question suggests using the Central Limit Theorem (CLT) to approximate this probability. Hmm, okay.First, I remember that the Poisson distribution is typically used for counts, like the number of events happening in a fixed interval. When ( lambda ) is large, the Poisson distribution can be approximated by a normal distribution. Since ( lambda = 5000 ) is pretty large, using the CLT makes sense here.So, if I approximate the Poisson distribution with a normal distribution, the mean ( mu ) would be equal to ( lambda ), which is 5000. The variance ( sigma^2 ) for a Poisson distribution is also equal to ( lambda ), so the standard deviation ( sigma ) is ( sqrt{5000} ). Let me calculate that:( sigma = sqrt{5000} approx 70.7107 )Okay, so the normal approximation is ( N(mu = 5000, sigma^2 = 5000) ).Now, I need to find the probability that ( V_i ) is between 4900 and 5100. Since we're dealing with a continuous approximation (normal distribution) for a discrete variable (Poisson), I should apply a continuity correction. That means I should adjust the boundaries by 0.5. So, instead of 4900 and 5100, I'll use 4899.5 and 5100.5.So, the probability we want is ( P(4899.5 leq V_i leq 5100.5) ).To find this probability, I can convert these values to z-scores using the formula:( z = frac{X - mu}{sigma} )Let me compute the z-scores for 4899.5 and 5100.5.First, for 4899.5:( z_1 = frac{4899.5 - 5000}{70.7107} approx frac{-100.5}{70.7107} approx -1.42 )Next, for 5100.5:( z_2 = frac{5100.5 - 5000}{70.7107} approx frac{100.5}{70.7107} approx 1.42 )So, the z-scores are approximately -1.42 and 1.42.Now, I need to find the probability that a standard normal variable Z is between -1.42 and 1.42. This is equivalent to:( P(-1.42 leq Z leq 1.42) = P(Z leq 1.42) - P(Z leq -1.42) )Looking up these z-scores in the standard normal distribution table or using a calculator.I remember that for a z-score of 1.42, the cumulative probability is about 0.9222. For a z-score of -1.42, the cumulative probability is 1 - 0.9222 = 0.0778.So, the probability between -1.42 and 1.42 is:0.9222 - 0.0778 = 0.8444Therefore, the approximate probability is 0.8444, or 84.44%.Wait, let me double-check the z-score calculations. 100.5 divided by 70.7107 is approximately 1.42? Let me compute 70.7107 * 1.42:70.7107 * 1.4 = 98.9949870.7107 * 0.02 = 1.414214Adding them together: 98.99498 + 1.414214 ≈ 100.4092, which is close to 100.5. So, yes, 1.42 is a reasonable approximation.Alternatively, if I use more precise calculations, maybe 1.42 is slightly off. Let me compute 100.5 / 70.7107 more accurately.100.5 / 70.7107 ≈ 1.4206So, actually, it's approximately 1.4206, which is about 1.42 when rounded to two decimal places.So, using 1.42 is fine.Therefore, the probability is approximately 84.44%.Alright, that seems solid.Moving on to part 2: The booktuber notices that for every 10,000 views, there's a 2% increase in their subscriber count. Their current subscriber base is 50,000, and only videos with views greater than 6000 contribute to this increase. I need to determine the expected increase in their subscriber count, using the properties of the Poisson distribution.Hmm, okay. So, first, let's parse this.They have 100 videos, each with views ( V_i ) ~ Poisson(5000). For each video, if ( V_i > 6000 ), then it contributes to an increase in subscribers. The increase is 2% per 10,000 views. So, for each such video, the increase is 2% times the number of 10,000 view increments above 6000.Wait, actually, the problem says: "for every 10,000 views, there is a 2% increase in their subscriber count due to engagement with these videos." So, it's 2% increase per 10,000 views. So, if a video has, say, 15,000 views, that's 1.5 times 10,000, so 1.5 * 2% = 3% increase.But the increase is per video? Or per total views across all videos? Wait, the problem says: "for every 10,000 views, there is a 2% increase in their subscriber count due to engagement with these videos." So, it's per 10,000 views across all videos? Or per video?Wait, the wording is a bit ambiguous. Let me read again: "for every 10,000 views, there is a 2% increase in their subscriber count due to engagement with these videos." So, it's a 2% increase for every 10,000 views, regardless of how many videos. So, if across all videos, there are, say, 100,000 views, that's 10 times 10,000, so 10 * 2% = 20% increase.But wait, the current subscriber base is 50,000. So, the increase is based on the number of views above 6000 per video? Or total views across all videos above 6000?Wait, the problem says: "only videos with views greater than 6000 contribute to this increase." So, each video that has more than 6000 views contributes 2% per 10,000 views to the subscriber count.So, for each video, if it has ( V_i ) views, and ( V_i > 6000 ), then the increase is 2% * (number of 10,000 views in ( V_i )).Wait, but the problem says "for every 10,000 views, there is a 2% increase in their subscriber count." So, it's 2% per 10,000 views, regardless of the video. So, if they have multiple videos, each contributing views, the total increase is 2% times (total views across all qualifying videos divided by 10,000).But the problem says "only videos with views greater than 6000 contribute to this increase." So, each video must have more than 6000 views to contribute, and the contribution is 2% per 10,000 views for that video.Wait, maybe it's per video. So, for each video with ( V_i > 6000 ), the increase is 2% * (number of 10,000 views in ( V_i )). So, if a video has 7000 views, that's 0.7 * 10,000, so 0.7 * 2% = 1.4% increase.But the problem says "for every 10,000 views, there is a 2% increase in their subscriber count." So, it's 2% per 10,000 views, regardless of how many videos. So, if they have multiple videos, each contributing views, the total increase is 2% times (total views across all qualifying videos divided by 10,000).But the problem is a bit ambiguous. Let me read again: "for every 10,000 views, there is a 2% increase in their subscriber count due to engagement with these videos. If their current subscriber base is 50,000, and only videos with views greater than 6000 contribute to this increase, determine the expected increase in their subscriber count."So, it's 2% increase per 10,000 views, and only videos with views >6000 contribute. So, the total number of views across all qualifying videos is summed, and for every 10,000 of that sum, they get a 2% increase.But the problem says "determine the expected increase in their subscriber count." So, the expected value of the increase.So, first, I need to find the expected number of views across all 100 videos where each video has views >6000. Then, for each 10,000 views in that total, they get a 2% increase.But wait, actually, the increase is 2% per 10,000 views, so the total increase is (Total Views / 10,000) * 2%.But the increase is applied to the current subscriber base of 50,000. So, the expected increase is 50,000 * (Total Views / 10,000) * 2%.Wait, no, that might not be correct. Let me think.If for every 10,000 views, the subscriber count increases by 2%, then the total increase is 2% * (Total Views / 10,000). So, the increase is 2% * (Total Views / 10,000). So, the expected increase would be 0.02 * (E[Total Views] / 10,000).But wait, the increase is multiplicative, right? Because a 2% increase is a multiplier of 1.02. But if you have multiple increments, it's 1.02^(Total Views / 10,000). But since the increase is small, maybe we can approximate it as additive? Or is it additive?Wait, the problem says "there is a 2% increase in their subscriber count due to engagement with these videos." So, for every 10,000 views, their subscriber count increases by 2%. So, if they have 10,000 views, their subscribers go up by 2%, which is 50,000 * 0.02 = 1,000. If they have 20,000 views, it's 2,000, etc.So, the increase is linear: 2% per 10,000 views. So, the total increase is (Total Views / 10,000) * 2% * 50,000.Wait, no. Wait, 2% increase is 2% of the current subscriber base. So, if they have 10,000 views, the increase is 2% of 50,000, which is 1,000. If they have 20,000 views, it's 2% * 50,000 * 2 = 2,000. So, the increase is (Total Views / 10,000) * 2% * 50,000.But wait, that would be (Total Views / 10,000) * 0.02 * 50,000.Alternatively, maybe it's 2% per 10,000 views, so for each 10,000 views, they get 2% more subscribers. So, the total increase is (Total Views / 10,000) * 0.02 * 50,000.But actually, if you have multiple increments, the increases compound. But since the problem says "there is a 2% increase in their subscriber count," it's probably additive, not multiplicative, because otherwise, it would be a 2% increase per 10,000 views, which would compound. But given the problem's wording, it's more likely additive.Wait, let me think again. If for every 10,000 views, the subscriber count increases by 2%, meaning that each 10,000 views adds 2% of the current subscriber base. So, if they have 10,000 views, their subscribers go up by 2%, so 50,000 * 0.02 = 1,000. If they have another 10,000 views, it's another 2% of 50,000, so another 1,000. So, it's additive, not multiplicative.Therefore, the total increase is (Total Views / 10,000) * 0.02 * 50,000.But wait, that would be (Total Views / 10,000) * 0.02 * 50,000. Simplifying, that's (Total Views) * 0.02 * 50,000 / 10,000 = Total Views * 0.02 * 5 = Total Views * 0.1.Wait, that can't be right because 0.02 * 50,000 is 1,000, and 1,000 per 10,000 views. So, per view, it's 1,000 / 10,000 = 0.1 per view. So, total increase is 0.1 * Total Views.But that seems high because 0.1 per view on 100 videos each with 5000 views on average would be 0.1 * 100 * 5000 = 50,000 increase, which would double their subscriber base. That seems too much.Wait, maybe I misinterpreted the problem. Let me read again: "for every 10,000 views, there is a 2% increase in their subscriber count due to engagement with these videos."So, 2% increase per 10,000 views. So, if they have 10,000 views, their subscriber count increases by 2%. If they have 20,000 views, it's 4%, and so on.But the problem says "determine the expected increase in their subscriber count." So, the expected increase is 2% * (E[Total Views] / 10,000). So, the expected increase is 0.02 * (E[Total Views] / 10,000).But wait, the increase is 2% of the current subscriber base per 10,000 views. So, if the current subscriber base is 50,000, then for each 10,000 views, the increase is 2% of 50,000, which is 1,000. So, the total increase is (Total Views / 10,000) * 1,000.But that would be 1,000 * (Total Views / 10,000) = 100 * Total Views.Wait, that can't be right because 100 * Total Views would be a huge number. Wait, no, 1,000 * (Total Views / 10,000) is 100 * Total Views. Wait, no, 1,000 * (Total Views / 10,000) is 100 * Total Views. Wait, that can't be.Wait, no, 1,000 * (Total Views / 10,000) is (1,000 / 10,000) * Total Views = 0.1 * Total Views. So, the increase is 0.1 * Total Views.Wait, but 0.1 * Total Views is 10% of Total Views. If Total Views is 100 * 5000 = 500,000, then 0.1 * 500,000 = 50,000. So, the increase would be 50,000, which would double their subscriber base. That seems too high.Wait, perhaps I'm overcomplicating. Let's think differently.The problem says: "for every 10,000 views, there is a 2% increase in their subscriber count." So, each 10,000 views adds 2% to the subscriber count. So, if they have V views, the increase is (V / 10,000) * 2%.But the current subscriber base is 50,000. So, the increase is 50,000 * (V / 10,000) * 0.02.Wait, that would be 50,000 * 0.02 * (V / 10,000) = 1,000 * (V / 10,000) = V / 10.So, the increase is V / 10. So, if V is 10,000, increase is 1,000. If V is 20,000, increase is 2,000, etc.But the problem is that only videos with views greater than 6000 contribute. So, we have 100 videos, each with views ( V_i ) ~ Poisson(5000). For each video, if ( V_i > 6000 ), then the views ( V_i ) contribute to the increase. So, the total views contributing to the increase is the sum of ( V_i ) for all ( V_i > 6000 ).Therefore, the total increase is (Total Views > 6000) / 10,000 * 2% * 50,000.Wait, but let me clarify:If the increase is 2% per 10,000 views, then for each 10,000 views, the subscriber count increases by 2%. So, the total increase is (Total Views / 10,000) * 2% of the current subscriber base.So, the formula is:Increase = (Total Views / 10,000) * 0.02 * 50,000Simplify:Increase = (Total Views / 10,000) * 1,000So, Increase = Total Views * 0.1But again, this seems high because if Total Views is 500,000, the increase would be 50,000, which is the same as the current subscriber base.Wait, perhaps the increase is 2% of the current subscriber base per 10,000 views. So, for each 10,000 views, they gain 2% of 50,000, which is 1,000. So, the total increase is (Total Views / 10,000) * 1,000.So, if Total Views is 500,000, then the increase is (500,000 / 10,000) * 1,000 = 50 * 1,000 = 50,000. So, again, doubling their subscriber base.But that seems extreme. Maybe the problem means that for every 10,000 views, the subscriber count increases by 2, meaning an absolute increase of 2, not 2%. That would make more sense, but the problem says 2%.Wait, the problem says: "for every 10,000 views, there is a 2% increase in their subscriber count." So, it's 2% increase, not 2 absolute. So, 2% of 50,000 is 1,000. So, for each 10,000 views, they gain 1,000 subscribers.Therefore, the total increase is (Total Views / 10,000) * 1,000.So, the expected increase is E[Total Views] / 10,000 * 1,000.But wait, only videos with views >6000 contribute. So, we need to find the expected total views from all videos where ( V_i > 6000 ).So, first, for each video, the expected views given that ( V_i > 6000 ). Then, multiply by the number of such videos.Wait, but actually, it's the sum over all videos of ( V_i ) if ( V_i > 6000 ). So, the expected total views is the sum over all videos of E[( V_i ) | ( V_i > 6000 )] * P(( V_i > 6000 )).Wait, no, more accurately, for each video, the expected contribution is E[( V_i ) * I(( V_i > 6000 ))], where I is the indicator function. So, the expected total views is 100 * E[( V_i ) * I(( V_i > 6000 ))].So, we need to compute E[( V ) | ( V > 6000 )] * P(( V > 6000 )).Wait, actually, E[( V ) * I(( V > 6000 ))] = E[( V ) | ( V > 6000 )] * P(( V > 6000 )).So, yes, that's correct.Therefore, for each video, the expected contribution is E[( V ) | ( V > 6000 )] * P(( V > 6000 )).So, first, let's compute P(( V > 6000 )) for a Poisson(5000) distribution.But calculating P(( V > 6000 )) for Poisson(5000) is going to be challenging because 6000 is quite far from the mean of 5000. Maybe we can approximate it using the normal distribution again.So, using the CLT, we can approximate the Poisson distribution with a normal distribution N(5000, 5000). Then, compute P(( V > 6000 )).So, let's compute the z-score for 6000.z = (6000 - 5000) / sqrt(5000) ≈ 1000 / 70.7107 ≈ 14.1421Wait, that's a z-score of about 14.14, which is extremely high. The probability of a normal variable being beyond 14 standard deviations is practically zero.Wait, that can't be right because 6000 is only 1000 away from 5000, but the standard deviation is about 70.71, so 1000 / 70.71 ≈ 14.14. So, yes, it's 14.14 standard deviations above the mean. The probability of that is effectively zero.But wait, in reality, the Poisson distribution is skewed, especially for large lambda, but even so, 6000 is way beyond the mean. So, the probability that ( V_i > 6000 ) is extremely small.But let me double-check. Maybe I made a mistake in the z-score.Wait, 6000 - 5000 = 1000. sqrt(5000) ≈ 70.71. So, 1000 / 70.71 ≈ 14.14. Yes, that's correct.So, the probability that a Poisson(5000) variable is greater than 6000 is approximately the probability that a normal variable is greater than 14.14, which is effectively zero.Therefore, the expected total views from videos with ( V_i > 6000 ) is approximately zero. Therefore, the expected increase in subscriber count is zero.But that seems counterintuitive. Maybe I made a mistake in the approach.Wait, perhaps I should use the Poisson distribution directly to compute P(( V > 6000 )). But calculating that exactly is computationally intensive because it's the sum from 6001 to infinity of (e^{-5000} * 5000^k) / k! which is not feasible by hand.Alternatively, maybe we can use the normal approximation with continuity correction.So, for P(( V > 6000 )), using continuity correction, we can compute P(( V geq 6001 )) ≈ P(Normal > 6000.5).So, z = (6000.5 - 5000) / sqrt(5000) ≈ 1000.5 / 70.7107 ≈ 14.1421 + 0.5 / 70.7107 ≈ 14.1421 + 0.00707 ≈ 14.1492.Still, the z-score is about 14.15, which is way beyond the standard normal table. The probability is effectively zero.Therefore, the probability that any video has more than 6000 views is practically zero. Therefore, the expected total views from such videos is zero, and thus the expected increase in subscribers is zero.But that seems odd because even though the probability is low, with 100 videos, maybe there's a chance that one or more exceed 6000.Wait, let me think. The probability that a single video has ( V_i > 6000 ) is extremely low, but with 100 videos, maybe the expected number of such videos is 100 * P(( V_i > 6000 )), which is still practically zero.Therefore, the expected total views is zero, leading to zero increase in subscribers.But that seems too definitive. Maybe I should compute the exact probability using the Poisson formula, but it's not feasible by hand.Alternatively, perhaps the problem expects us to use the Poisson distribution's properties without approximating, but I don't see a straightforward way.Wait, another approach: For a Poisson distribution, the probability that ( V > 6000 ) is equal to 1 - P(( V leq 6000 )). But calculating P(( V leq 6000 )) for Poisson(5000) is still difficult.Alternatively, maybe we can use the normal approximation with continuity correction, but as we saw, the z-score is so high that the probability is negligible.Therefore, perhaps the expected increase is zero.But let me think again. Maybe the problem expects us to calculate the expected number of videos with ( V_i > 6000 ), and then for each such video, calculate the expected views above 6000, and then compute the increase.Wait, that's a different approach. Let's try that.So, first, find the expected number of videos with ( V_i > 6000 ). That would be 100 * P(( V_i > 6000 )).But as we saw, P(( V_i > 6000 )) is practically zero, so the expected number is zero.Therefore, the expected total views from such videos is zero, leading to zero increase.Alternatively, maybe the problem expects us to calculate the expected value of ( V_i ) given ( V_i > 6000 ), and then multiply by P(( V_i > 6000 )) and then by 100, and then compute the increase.But since P(( V_i > 6000 )) is practically zero, the expected value would still be zero.Alternatively, maybe the problem is expecting us to use the memoryless property or something else, but I don't think that applies here.Wait, another thought: Maybe the problem is expecting us to calculate the expected number of views above 6000 for each video, and then sum that over all videos, and then compute the increase based on that.So, for each video, the expected excess views above 6000 is E[( V_i - 6000 ) | ( V_i > 6000 )] * P(( V_i > 6000 )).But again, since P(( V_i > 6000 )) is practically zero, the expected excess is zero.Therefore, the expected total excess views is zero, leading to zero increase.But this seems too straightforward. Maybe I'm missing something.Wait, perhaps the problem is expecting us to calculate the expected number of views for each video, regardless of the threshold, and then apply the 2% increase per 10,000 views. But no, the problem specifies that only videos with views greater than 6000 contribute.Alternatively, maybe the problem is expecting us to calculate the expected number of views above 6000 for each video, and then sum that over all videos, and then compute the increase based on that.But again, since the probability is so low, the expected value is negligible.Alternatively, maybe the problem is expecting us to use the fact that for Poisson distribution, E[( V )] = 5000, and Var(( V )) = 5000, and then model the total views as a normal distribution with mean 100*5000 = 500,000 and variance 100*5000 = 500,000, so standard deviation sqrt(500,000) ≈ 707.107.Then, compute P(Total Views > 6000*100 = 600,000). Wait, but that's not the case because each video is independent, and we're only considering videos with ( V_i > 6000 ).Wait, no, that approach is not correct because we're summing only the views from videos that exceed 6000.This is getting complicated. Maybe the problem expects us to recognize that the probability is so low that the expected increase is negligible, effectively zero.Alternatively, perhaps the problem expects us to use the Poisson distribution's properties to calculate the expected value of ( V_i ) given ( V_i > 6000 ), but I don't recall a formula for that off the top of my head.Wait, for a Poisson distribution, the expected value given that ( V geq k ) is ( lambda + ) something, but I don't remember the exact formula.Alternatively, maybe we can use the fact that for Poisson, E[( V ) | ( V geq k )] = ( lambda + ) something, but I'm not sure.Alternatively, perhaps we can use the conditional expectation formula:E[( V ) | ( V > 6000 )] = (E[( V ) * I(( V > 6000 ))]) / P(( V > 6000 ))But since both numerator and denominator are difficult to compute, perhaps we can approximate.But given that P(( V > 6000 )) is practically zero, the expected value might be very large, but multiplied by a near-zero probability, the expected contribution per video is still negligible.Therefore, the expected total views from all videos with ( V_i > 6000 ) is approximately zero, leading to zero expected increase in subscribers.But I'm not entirely confident about this conclusion because it's possible that with 100 videos, even a small probability could lead to a non-zero expectation.Wait, let's think in terms of linearity of expectation. The expected total views from videos with ( V_i > 6000 ) is the sum over all videos of E[( V_i ) * I(( V_i > 6000 ))].So, for each video, E[( V_i ) * I(( V_i > 6000 ))] = E[( V_i ) | ( V_i > 6000 )] * P(( V_i > 6000 )).But without knowing E[( V_i ) | ( V_i > 6000 )], it's hard to compute.Alternatively, perhaps we can approximate E[( V_i ) | ( V_i > 6000 )] as 6000 + something, but I don't know.Alternatively, maybe we can use the normal approximation for the conditional expectation.So, if we approximate ( V_i ) as N(5000, 5000), then E[( V_i ) | ( V_i > 6000 )] can be approximated as the mean of the truncated normal distribution above 6000.The formula for the mean of a truncated normal distribution is:E[X | X > a] = μ + σ * φ((a - μ)/σ) / (1 - Φ((a - μ)/σ))Where φ is the standard normal PDF and Φ is the standard normal CDF.So, let's compute that.Given μ = 5000, σ = sqrt(5000) ≈ 70.7107.a = 6000.Compute z = (6000 - 5000) / 70.7107 ≈ 14.1421.So, φ(z) is the standard normal PDF at z = 14.1421, which is effectively zero because the PDF at such a high z is negligible.Similarly, Φ(z) is practically 1, so 1 - Φ(z) is practically zero.Therefore, the ratio φ(z) / (1 - Φ(z)) is approximately zero / zero, which is indeterminate, but in reality, for such high z, the ratio approaches zero because φ(z) decreases much faster than 1 - Φ(z).Therefore, E[X | X > a] ≈ μ + σ * 0 = μ = 5000.But that can't be right because if we condition on X > a, where a is much higher than μ, the conditional expectation should be higher than μ.Wait, perhaps the approximation breaks down for such extreme z-scores.Alternatively, maybe we can use the fact that for z >> 0, the conditional expectation E[X | X > a] ≈ a + σ / (2φ(z)).But I'm not sure about that.Alternatively, perhaps it's better to accept that for such a high z-score, the conditional expectation is approximately a + σ / (2φ(z)), but since φ(z) is negligible, this would be a very large number, but multiplied by the near-zero probability, the expected contribution per video is still negligible.Therefore, perhaps the expected total views is negligible, leading to zero expected increase.Alternatively, maybe the problem expects us to recognize that the probability is so low that the expected increase is zero, and thus the answer is zero.But I'm not entirely sure. Maybe I should proceed with the calculation as per the normal approximation.So, for each video, the expected contribution is E[( V_i ) | ( V_i > 6000 )] * P(( V_i > 6000 )).Using the normal approximation, E[( V_i ) | ( V_i > 6000 )] ≈ μ + σ * φ(z) / (1 - Φ(z)).But as z is very large, φ(z) is negligible, so E[( V_i ) | ( V_i > 6000 )] ≈ μ + 0 = μ = 5000.But that seems contradictory because if we condition on ( V_i > 6000 ), the expectation should be higher than 5000.Wait, perhaps the formula is different. Let me recall the formula for the conditional expectation of a truncated normal distribution.Yes, the formula is:E[X | X > a] = μ + σ * φ((a - μ)/σ) / (1 - Φ((a - μ)/σ))So, with μ = 5000, σ ≈ 70.7107, a = 6000.z = (6000 - 5000) / 70.7107 ≈ 14.1421.φ(z) is the standard normal PDF at z = 14.1421, which is approximately zero.Similarly, 1 - Φ(z) is approximately zero.But we can use the approximation for the ratio φ(z) / (1 - Φ(z)) ≈ z for large z.Wait, actually, for large z, 1 - Φ(z) ≈ φ(z) / z.Therefore, φ(z) / (1 - Φ(z)) ≈ z.So, E[X | X > a] ≈ μ + σ * z.Wait, z = (a - μ)/σ, so E[X | X > a] ≈ μ + σ * (a - μ)/σ = a.Wait, that can't be right because it would imply E[X | X > a] = a, which is not true.Wait, let me check the approximation.For large z, 1 - Φ(z) ≈ φ(z) / z.Therefore, φ(z) / (1 - Φ(z)) ≈ z.So, E[X | X > a] ≈ μ + σ * z.But z = (a - μ)/σ, so E[X | X > a] ≈ μ + σ * (a - μ)/σ = μ + (a - μ) = a.Wait, that suggests E[X | X > a] ≈ a for large z, which is not correct because the expectation should be higher than a.Wait, perhaps the approximation is different.Wait, another approach: For large z, the conditional expectation E[X | X > a] ≈ a + σ / (2φ(z)).But I'm not sure.Alternatively, perhaps it's better to accept that for such a high z-score, the conditional expectation is approximately a + σ / (2φ(z)), but since φ(z) is negligible, this term is huge, but multiplied by the near-zero probability, the expected contribution is still negligible.Therefore, perhaps the expected total views is approximately zero.Given that, the expected increase in subscribers is zero.But I'm not entirely confident. Maybe the problem expects us to recognize that the probability is negligible and thus the expected increase is zero.Alternatively, perhaps the problem expects us to use the Poisson distribution's properties to calculate the expected value.Wait, another thought: For a Poisson distribution, the expected value of V given V > k is equal to λ + something, but I don't recall the exact formula.Alternatively, perhaps we can use the formula for the expected value of a truncated Poisson distribution.The expected value of a Poisson distribution truncated at k is:E[V | V > k] = (λ^{k+1} / k!) * e^{-λ} / (1 - Γ(k+1, λ)/k!) )But this is complicated and not feasible by hand.Alternatively, perhaps we can use the fact that for Poisson, E[V | V > k] = λ + something, but I don't remember.Alternatively, perhaps we can use the recursion formula for Poisson probabilities.But given the time constraints, maybe it's better to proceed with the conclusion that the expected increase is zero.Therefore, the expected increase in subscriber count is zero.But wait, let me think again. If the probability is so low, but with 100 videos, maybe the expected number of videos with ( V_i > 6000 ) is 100 * P(( V_i > 6000 )), which is still practically zero.Therefore, the expected total views is zero, leading to zero increase.Alternatively, maybe the problem expects us to calculate the expected value of ( V_i ) for each video, which is 5000, and then compute the increase based on that, but the problem specifies that only videos with ( V_i > 6000 ) contribute, so that approach is invalid.Therefore, I think the expected increase is zero.But I'm not entirely sure. Maybe I should look for another approach.Wait, another idea: Maybe the problem expects us to calculate the expected number of views above 6000 for each video, which is E[( V_i - 6000 ) | ( V_i > 6000 )] * P(( V_i > 6000 )), and then sum over all videos.But again, since P(( V_i > 6000 )) is practically zero, the expected value is zero.Therefore, the expected increase is zero.Alright, I think I have to conclude that the expected increase is zero.So, summarizing:1. The probability that a video has between 4900 and 5100 views is approximately 84.44%.2. The expected increase in subscriber count is zero.But wait, the second part seems too definitive. Maybe I should express it as a very small number, but practically zero.Alternatively, perhaps the problem expects us to use the Poisson distribution's properties without approximating, but I don't see a way to compute it exactly.Alternatively, maybe the problem expects us to recognize that the expected number of views above 6000 is zero, leading to zero increase.Therefore, I think the answer is zero.Final Answer1. The probability is approximately boxed{0.8444}.2. The expected increase in subscriber count is boxed{0}.</think>

🧠:<think>Okay, so I need to help Alex figure out their optimal daily routine based on their health-conscious goals. Let me start by understanding the problem step by step.First, Alex has a daily caloric intake ( C ) that should be between 1800 and 2200 calories. That's a good range, not too low, not too high. Then, they have a model for caloric expenditure based on physical activity hours ( H ). The expenditure function is given by ( E(H) = 400H + 0.1H^3 ). The first question is asking for the range of ( H ) such that the net caloric balance, which is intake minus expenditure, stays between 1500 and 2000 calories. So, mathematically, this means:( 1500 leq C - E(H) leq 2000 )But since ( C ) is between 1800 and 2200, we can write:( 1500 leq 1800 - E(H) leq 2000 ) and ( 1500 leq 2200 - E(H) leq 2000 )Wait, actually, maybe I should consider the net balance for both the minimum and maximum caloric intake. So, for the net balance to be at least 1500, when ( C ) is at its minimum (1800), we have:( 1800 - E(H) geq 1500 )Similarly, for the net balance to be at most 2000, when ( C ) is at its maximum (2200), we have:( 2200 - E(H) leq 2000 )So, these two inequalities will give us the range for ( E(H) ). Let me compute these:First inequality: ( 1800 - E(H) geq 1500 ) Subtract 1800: ( -E(H) geq -300 ) Multiply both sides by -1 (and reverse inequality): ( E(H) leq 300 )Second inequality: ( 2200 - E(H) leq 2000 ) Subtract 2200: ( -E(H) leq -200 ) Multiply both sides by -1 (and reverse inequality): ( E(H) geq 200 )So, combining these, ( 200 leq E(H) leq 300 ). Therefore, we need to find the values of ( H ) such that ( 200 leq 400H + 0.1H^3 leq 300 ).Now, let's solve ( 400H + 0.1H^3 = 200 ) and ( 400H + 0.1H^3 = 300 ) to find the range of ( H ).Starting with the lower bound: ( 0.1H^3 + 400H - 200 = 0 )This is a cubic equation. It might be a bit tricky to solve algebraically, so maybe I can use numerical methods or graphing to approximate the solution.Similarly, for the upper bound: ( 0.1H^3 + 400H - 300 = 0 )Same approach here.Alternatively, since ( H ) is in hours, and physical activity hours are likely to be a small number, maybe I can test integer values to approximate.Let me try plugging in H=0.5:Lower equation: 0.1*(0.5)^3 + 400*(0.5) = 0.1*(0.125) + 200 = 0.0125 + 200 = 200.0125. That's just above 200. So H=0.5 gives E(H)=200.0125.Similarly, for H=0.5, it's just over 200. Let me check H=0.4:0.1*(0.4)^3 + 400*(0.4) = 0.1*(0.064) + 160 = 0.0064 + 160 = 160.0064. That's too low.So, between H=0.4 and H=0.5, E(H) crosses 200. Since at H=0.5, it's just over 200, maybe H≈0.5 is the lower bound.Now, for the upper bound, E(H)=300.Let me try H=0.6:0.1*(0.6)^3 + 400*(0.6) = 0.1*(0.216) + 240 = 0.0216 + 240 = 240.0216. Still below 300.H=0.7:0.1*(0.343) + 400*(0.7) = 0.0343 + 280 = 280.0343. Still below 300.H=0.8:0.1*(0.512) + 400*(0.8) = 0.0512 + 320 = 320.0512. That's above 300.So, between H=0.7 and H=0.8, E(H) crosses 300.Let me do a linear approximation between H=0.7 and H=0.8.At H=0.7, E=280.0343 At H=0.8, E=320.0512We need E=300. The difference between 280.0343 and 320.0512 is about 39.0169 over 0.1 hours.So, to reach 300 from 280.0343, we need 19.9657.So, fraction = 19.9657 / 39.0169 ≈ 0.511.Therefore, H ≈ 0.7 + 0.511*(0.1) ≈ 0.7 + 0.0511 ≈ 0.7511 hours.So, approximately 0.75 hours.Therefore, the range of H is approximately from 0.5 to 0.75 hours.But wait, let me double-check with H=0.75:E(H) = 0.1*(0.75)^3 + 400*(0.75) = 0.1*(0.421875) + 300 = 0.0421875 + 300 = 300.0421875.Yes, that's just over 300. So, H≈0.75 is the upper bound.Similarly, for the lower bound, H=0.5 gives E≈200.0125, which is just over 200.So, the range of H is approximately 0.5 to 0.75 hours.But let me check if H can be as low as 0.5 hours. Is that feasible? 0.5 hours is 30 minutes. That seems reasonable for physical activity.Similarly, 0.75 hours is 45 minutes. So, Alex needs to spend between 30 and 45 minutes on physical activities each day to keep their net caloric balance between 1500 and 2000 calories.Wait, hold on. Let me think again.The net caloric balance is intake minus expenditure. So, if Alex's intake is between 1800 and 2200, and expenditure is E(H), then:For the net balance to be at least 1500, when C is 1800:1800 - E(H) ≥ 1500 E(H) ≤ 300And for the net balance to be at most 2000, when C is 2200:2200 - E(H) ≤ 2000 E(H) ≥ 200So, E(H) must be between 200 and 300. Therefore, H must be such that E(H) is in [200,300]. So, H must be between approximately 0.5 and 0.75 hours.But wait, let me confirm with H=0.5:E(0.5) = 400*0.5 + 0.1*(0.5)^3 = 200 + 0.0125 = 200.0125, which is just above 200.Similarly, H=0.75:E(0.75) = 400*0.75 + 0.1*(0.75)^3 = 300 + 0.0421875 ≈ 300.0422, which is just above 300.So, the range of H is approximately 0.5 ≤ H ≤ 0.75 hours.But let me check if H can be less than 0.5 or more than 0.75.If H=0.4, E=160.0064, which would make the net balance when C=1800: 1800 - 160.0064 ≈ 1639.99, which is above 1500. But wait, the net balance needs to be at least 1500, so 1639 is acceptable. But the problem is, the net balance also needs to be at most 2000 when C=2200.Wait, no, the net balance is intake minus expenditure. So, if E(H) is too low, when C is high, the net balance could be too high.Wait, let me clarify:The net caloric balance must be between 1500 and 2000. So, for any C between 1800 and 2200, C - E(H) must be between 1500 and 2000.Wait, that might not be correct. Because C can vary, but E(H) is fixed for a given H. So, actually, for a given H, the net balance can vary depending on C.But the problem says "the net caloric balance (intake minus expenditure) between 1500 and 2000 calories."Wait, maybe I misinterpreted. Perhaps it's that for the given H, the net balance must be between 1500 and 2000 regardless of C? Or is it that for the given C, the net balance must be between 1500 and 2000?Wait, the problem says: "the caloric expenditure E(H) keeps Alex's net caloric balance (intake minus expenditure) between 1500 and 2000 calories."So, it's for a given H, the net balance (C - E(H)) must be between 1500 and 2000. But C itself is between 1800 and 2200.So, to ensure that for all C in [1800,2200], C - E(H) is in [1500,2000].Wait, that would mean:For the minimum C (1800), 1800 - E(H) ≥ 1500 => E(H) ≤ 300And for the maximum C (2200), 2200 - E(H) ≤ 2000 => E(H) ≥ 200So, combining these, E(H) must be between 200 and 300.Therefore, H must satisfy 200 ≤ 400H + 0.1H^3 ≤ 300.So, solving for H in this inequality.We already found that H≈0.5 gives E≈200.01 and H≈0.75 gives E≈300.04.Therefore, the range of H is approximately 0.5 ≤ H ≤ 0.75 hours.But let me check H=0.6:E=400*0.6 + 0.1*(0.6)^3=240 + 0.0216=240.0216, which is between 200 and 300.Similarly, H=0.7:E=400*0.7 + 0.1*(0.7)^3=280 + 0.0343=280.0343, still between 200 and 300.So, the range is from approximately 0.5 to 0.75 hours.But let me see if there's a more precise way to solve 400H + 0.1H^3 = 200 and 300.For the lower bound:0.1H^3 + 400H - 200 = 0Let me try H=0.5:0.1*(0.125) + 400*(0.5) - 200 = 0.0125 + 200 - 200 = 0.0125 >0H=0.49:0.1*(0.49)^3 + 400*(0.49) - 2000.1*(0.117649) + 196 - 200 = 0.0117649 + 196 -200 ≈ -3.9882351So, between H=0.49 and H=0.5, the function crosses zero.Using linear approximation:At H=0.49, f=-3.9882351 At H=0.5, f=0.0125The difference is about 3.9882351 + 0.0125 = 4.0007351 over 0.01 hours.We need to find H where f=0.So, the zero crossing is at H=0.49 + (0 - (-3.9882351))/4.0007351 *0.01≈ 0.49 + (3.9882351)/4.0007351 *0.01≈ 0.49 + 0.9969 *0.01 ≈ 0.49 + 0.009969 ≈ 0.499969 ≈ 0.5 hours.So, H≈0.5 hours is the lower bound.Similarly, for the upper bound:0.1H^3 + 400H - 300 =0At H=0.75:0.1*(0.421875) + 400*(0.75) -300=0.0421875 +300 -300=0.0421875>0At H=0.74:0.1*(0.74)^3 +400*(0.74) -3000.1*(0.405224) +296 -300≈0.0405224 +296 -300≈-3.9594776So, between H=0.74 and H=0.75, f crosses zero.Using linear approximation:At H=0.74, f≈-3.9594776 At H=0.75, f≈0.0421875Difference≈3.9594776 +0.0421875≈4.0016651 over 0.01 hours.Zero crossing at H=0.74 + (0 - (-3.9594776))/4.0016651 *0.01≈0.74 + (3.9594776)/4.0016651 *0.01≈0.74 + 0.9896 *0.01≈0.74 +0.009896≈0.749896≈0.75 hours.So, H≈0.75 hours is the upper bound.Therefore, the range of H is approximately 0.5 to 0.75 hours.Now, moving on to the second question.Alex decides to vary their physical activity hours H according to a sinusoidal function over a week: H(t) = 2 + sin(πt/7), where t is the day of the week from 0 to 7.We need to calculate the total caloric expenditure for the week using this function.So, total expenditure is the sum of E(H(t)) for t=0 to t=6 (since t=7 would be the same as t=0, as sin is periodic with period 2π, and π*7/7=π, so sin(π)=0, so H(7)=2+0=2, same as H(0)=2+0=2.Wait, actually, t is from 0 to 7, but day 7 would be the next day, so maybe we need to include t=7 as well? Or is it from t=0 to t=6?The problem says "over a week, t is the day of the week (from 0 to 7)". So, t=0 to t=7, inclusive. That would be 8 days, but a week is 7 days. Hmm, maybe t=0 to t=6, which is 7 days.But the function is defined for t from 0 to 7, so perhaps we need to integrate over t=0 to t=7.Wait, the problem says "calculate the total caloric expenditure for the week using this function." So, since a week has 7 days, t=0 to t=6, but the function is defined up to t=7. Maybe it's better to integrate over t=0 to t=7, which would cover the entire period, even though it's 8 points.But actually, since H(t) is a continuous function over the week, maybe we should integrate over the interval [0,7] to find the total expenditure.Wait, but the function H(t) is given as 2 + sin(πt/7). So, over t=0 to t=7, the function completes half a period, because the period of sin(πt/7) is 14, so from 0 to 7 is half a period.Wait, no, the period of sin(k t) is 2π/k. Here, k=π/7, so period is 2π/(π/7)=14. So, from t=0 to t=7 is half a period.Therefore, the function H(t) goes from 2 + sin(0)=2 to 2 + sin(π)=2 +0=2, but in between, it reaches a maximum of 2 +1=3 at t=3.5.So, the function is symmetric around t=3.5.But to find the total caloric expenditure, we need to compute the integral of E(H(t)) over t from 0 to 7.Because E(H) is given as a function of H, which is a function of t. So, total expenditure is ∫₀⁷ E(H(t)) dt.Given E(H) = 400H + 0.1H³, and H(t)=2 + sin(πt/7).So, total expenditure = ∫₀⁷ [400*(2 + sin(πt/7)) + 0.1*(2 + sin(πt/7))³] dtLet me expand this:Total = ∫₀⁷ [800 + 400 sin(πt/7) + 0.1*(8 + 12 sin(πt/7) + 6 sin²(πt/7) + sin³(πt/7))] dtWait, let me compute (2 + sin x)^3:(2 + sin x)^3 = 8 + 12 sin x + 6 sin²x + sin³xSo, substituting back:Total = ∫₀⁷ [800 + 400 sin(πt/7) + 0.1*(8 + 12 sin(πt/7) + 6 sin²(πt/7) + sin³(πt/7))] dtNow, let's distribute the 0.1:= ∫₀⁷ [800 + 400 sin(πt/7) + 0.8 + 1.2 sin(πt/7) + 0.6 sin²(πt/7) + 0.1 sin³(πt/7)] dtCombine like terms:Constant terms: 800 + 0.8 = 800.8Sin terms: 400 sin(πt/7) + 1.2 sin(πt/7) = 401.2 sin(πt/7)Sin² term: 0.6 sin²(πt/7)Sin³ term: 0.1 sin³(πt/7)So, total integral becomes:∫₀⁷ [800.8 + 401.2 sin(πt/7) + 0.6 sin²(πt/7) + 0.1 sin³(πt/7)] dtNow, we can split this into four separate integrals:Total = ∫₀⁷ 800.8 dt + ∫₀⁷ 401.2 sin(πt/7) dt + ∫₀⁷ 0.6 sin²(πt/7) dt + ∫₀⁷ 0.1 sin³(πt/7) dtLet's compute each integral separately.First integral: ∫₀⁷ 800.8 dt = 800.8 * (7 - 0) = 800.8 *7 = 5605.6Second integral: ∫₀⁷ 401.2 sin(πt/7) dtLet me compute ∫ sin(πt/7) dt.Let u = πt/7, so du = π/7 dt, dt = 7/π duSo, ∫ sin(u) * (7/π) du = -7/π cos(u) + CTherefore, ∫₀⁷ sin(πt/7) dt = [-7/π cos(πt/7)] from 0 to7= [-7/π cos(π) + 7/π cos(0)] = [-7/π (-1) + 7/π (1)] = 7/π +7/π =14/πSo, the second integral is 401.2 * (14/π) ≈ 401.2 *4.4429≈ let's compute 401.2*4.4429First, 400*4.4429=1777.161.2*4.4429≈5.3315Total≈1777.16 +5.3315≈1782.4915Third integral: ∫₀⁷ 0.6 sin²(πt/7) dtWe can use the identity sin²x = (1 - cos(2x))/2So, ∫ sin²(πt/7) dt = ∫ (1 - cos(2πt/7))/2 dt = (1/2)∫1 dt - (1/2)∫cos(2πt/7) dtCompute from 0 to7:First term: (1/2)*(7 -0)=3.5Second term: -(1/2)*(7/(2π)) sin(2πt/7) evaluated from 0 to7At t=7: sin(2π*7/7)=sin(2π)=0 At t=0: sin(0)=0 So, the second term is zero.Therefore, ∫₀⁷ sin²(πt/7) dt =3.5So, the third integral is 0.6 *3.5=2.1Fourth integral: ∫₀⁷ 0.1 sin³(πt/7) dtThis one is trickier. Let's recall that sin³x can be expressed using the identity:sin³x = (3 sinx - sin3x)/4So, ∫ sin³x dx = ∫ (3 sinx - sin3x)/4 dx = (-3/4 cosx + 1/12 cos3x) + CSo, applying this to our integral:Let u=πt/7, so du=π/7 dt, dt=7/π duSo, ∫ sin³(πt/7) dt = ∫ sin³u * (7/π) du = (7/π) ∫ sin³u du = (7/π)[ (-3/4 cosu + 1/12 cos3u) ] + CTherefore, ∫₀⁷ sin³(πt/7) dt = (7/π)[ (-3/4 cos(πt/7) + 1/12 cos(3πt/7) ) ] from 0 to7Evaluate at t=7:cos(π*7/7)=cos(π)= -1 cos(3π*7/7)=cos(3π)= -1Evaluate at t=0:cos(0)=1 cos(0)=1So, plugging in:= (7/π)[ (-3/4*(-1) +1/12*(-1)) - (-3/4*(1) +1/12*(1)) ]Simplify:= (7/π)[ (3/4 -1/12) - (-3/4 +1/12) ]Compute inside the brackets:First part: 3/4 -1/12 = 9/12 -1/12=8/12=2/3 Second part: -3/4 +1/12= -9/12 +1/12= -8/12= -2/3So, subtracting: (2/3) - (-2/3)=2/3 +2/3=4/3Therefore, ∫₀⁷ sin³(πt/7) dt= (7/π)*(4/3)=28/(3π)≈28/(9.4248)≈2.97So, the fourth integral is 0.1 *2.97≈0.297Now, summing up all four integrals:First:5605.6 Second:≈1782.4915 Third:2.1 Fourth:≈0.297Total≈5605.6 +1782.4915 +2.1 +0.297≈Let me add step by step:5605.6 +1782.4915=7388.0915 7388.0915 +2.1=7390.1915 7390.1915 +0.297≈7390.4885So, approximately 7390.49 calories.But let me check the calculations again, especially the second integral.Wait, the second integral was 401.2*(14/π). Let me compute 14/π≈4.442882908So, 401.2*4.442882908≈Compute 400*4.442882908=1777.153163 1.2*4.442882908≈5.3314595 Total≈1777.153163 +5.3314595≈1782.484623So, more accurately, 1782.4846Third integral was 2.1, fourth was≈0.297So, total:5605.6 +1782.4846=7388.0846 7388.0846 +2.1=7390.1846 7390.1846 +0.297≈7390.4816So, approximately 7390.48 calories.But let me check if I made a mistake in the fourth integral.Wait, when I computed ∫ sin³(πt/7) dt from 0 to7, I got 28/(3π)≈2.97. Then multiplied by 0.1 gives≈0.297.Yes, that seems correct.So, the total caloric expenditure for the week is approximately 7390.48 calories.But let me see if I can express this more precisely.Alternatively, maybe I can keep it symbolic.Wait, let me re-express the total:Total =5605.6 + (401.2)*(14/π) +2.1 +0.1*(28/(3π))Compute each term:5605.6 is exact.401.2*(14/π)= (401.2*14)/π=5616.8/π≈5616.8/3.1415926536≈1787.14Wait, wait, earlier I computed 401.2*(14/π)=401.2*4.4429≈1782.49, but 401.2*14=5616.8, and 5616.8/π≈1787.14Wait, that's conflicting with my previous calculation.Wait, no, 14/π≈4.4429, so 401.2*4.4429≈1782.49But 401.2*14=5616.8, and 5616.8/π≈1787.14Wait, which is correct?Wait, 401.2*(14/π)= (401.2*14)/π=5616.8/π≈1787.14But earlier, I computed 401.2*4.4429≈1782.49Wait, 4.4429*401.2=?Let me compute 4*401.2=1604.8 0.4429*401.2≈0.4*401.2=160.48; 0.0429*401.2≈17.20 Total≈160.48+17.20=177.68 So, total≈1604.8 +177.68≈1782.48But 5616.8/π≈5616.8/3.1416≈1787.14Wait, so which one is correct?Wait, 401.2*(14/π)=401.2*4.4429≈1782.48But 401.2*14=5616.8, and 5616.8/π≈1787.14Wait, that's inconsistent. Wait, no, 401.2*(14/π)= (401.2*14)/π=5616.8/π≈1787.14But earlier, I thought 401.2*(14/π)=401.2*4.4429≈1782.48Wait, that's a mistake. Because 14/π≈4.4429, so 401.2*4.4429≈1782.48But 401.2*14=5616.8, and 5616.8/π≈1787.14Wait, so which is correct?Wait, 401.2*(14/π)=401.2*4.4429≈1782.48But 401.2*14=5616.8, and 5616.8/π≈1787.14Wait, but 401.2*(14/π)= (401.2*14)/π=5616.8/π≈1787.14So, I think I made a mistake earlier. The correct value is≈1787.14Wait, let me compute 401.2*(14/π):14/π≈4.442882908401.2*4.442882908≈Compute 400*4.442882908=1777.153163 1.2*4.442882908≈5.3314595 Total≈1777.153163 +5.3314595≈1782.484623Wait, so it's≈1782.48But 401.2*14=5616.8, and 5616.8/π≈1787.14Wait, that's conflicting.Wait, no, 401.2*(14/π)= (401.2*14)/π=5616.8/π≈1787.14But 401.2*(14/π)=401.2*4.4429≈1782.48Wait, that can't be. There's a miscalculation here.Wait, no, 14/π≈4.4429, so 401.2*4.4429≈1782.48But 401.2*14=5616.8, and 5616.8/π≈1787.14Wait, so which is correct? It must be that 401.2*(14/π)=5616.8/π≈1787.14But when I compute 401.2*4.4429≈1782.48, that's incorrect because 401.2*(14/π)= (401.2*14)/π=5616.8/π≈1787.14Wait, I think I confused the multiplication order.Yes, because 401.2*(14/π)= (401.2*14)/π=5616.8/π≈1787.14So, the correct value is≈1787.14Therefore, my earlier calculation was wrong. The second integral should be≈1787.14So, let me recalculate the total:First integral:5605.6 Second integral:≈1787.14 Third integral:2.1 Fourth integral:≈0.297Total≈5605.6 +1787.14 +2.1 +0.297≈5605.6 +1787.14=7392.74 7392.74 +2.1=7394.84 7394.84 +0.297≈7395.137So, approximately 7395.14 calories.But wait, let me compute 5616.8/π more accurately.π≈3.14159265365616.8 /3.1415926536≈Compute 3.1415926536*1787=?3.1415926536*1787≈Compute 3*1787=5361 0.1415926536*1787≈253.0 Total≈5361 +253=5614So, 3.1415926536*1787≈5614But 5616.8 is slightly more, so 1787 + (5616.8 -5614)/3.1415926536≈1787 +2.8/3.1416≈1787 +0.89≈1787.89So, 5616.8/π≈1787.89Therefore, the second integral is≈1787.89So, total≈5605.6 +1787.89 +2.1 +0.297≈5605.6 +1787.89=7393.49 7393.49 +2.1=7395.59 7395.59 +0.297≈7395.887So, approximately 7395.89 calories.But let me check the fourth integral again.Earlier, I computed ∫ sin³(πt/7) dt from 0 to7=28/(3π)≈2.97But let me verify:We had:∫ sin³(πt/7) dt from 0 to7= (7/π)*(4/3)=28/(3π)≈28/9.4248≈2.97Yes, that's correct.So, 0.1*2.97≈0.297So, the total is≈5605.6 +1787.89 +2.1 +0.297≈7395.89But let me compute it more accurately.5605.6 +1787.89=7393.49 7393.49 +2.1=7395.59 7395.59 +0.297=7395.887So,≈7395.89 calories.But let me see if I can express this in terms of π.The total expenditure is:5605.6 + (401.2*14)/π +2.1 +0.1*(28/(3π))=5605.6 + (5616.8)/π +2.1 +2.8/(3π)=5605.6 +2.1 + (5616.8 +2.8/3)/π=5607.7 + (5616.8 +0.9333)/π=5607.7 +5617.7333/πCompute 5617.7333/π≈5617.7333/3.1415926536≈1788.5So, total≈5607.7 +1788.5≈7396.2Which is close to our previous approximation.Therefore, the total caloric expenditure for the week is approximately 7396 calories.But let me see if I can express it more precisely.Alternatively, maybe I can keep it in terms of π.Total =5605.6 + (401.2*14)/π +2.1 +0.1*(28/(3π))=5605.6 + (5616.8)/π +2.1 +2.8/(3π)=5605.6 +2.1 + (5616.8 +2.8/3)/π=5607.7 + (5616.8 +0.9333)/π=5607.7 +5617.7333/πSo, Total=5607.7 +5617.7333/πBut if we want to write it as a single expression, it's:Total=5607.7 + (5617.7333)/πBut for the purpose of the answer, probably approximate to the nearest whole number.So,≈7396 calories.But let me check if I made any mistake in the integrals.Wait, the first integral was ∫800.8 dt from0 to7=800.8*7=5605.6, correct.Second integral: ∫401.2 sin(πt/7) dt from0 to7=401.2*(14/π)=5616.8/π≈1787.14, correct.Third integral: ∫0.6 sin²(πt/7) dt from0 to7=0.6*3.5=2.1, correct.Fourth integral: ∫0.1 sin³(πt/7) dt from0 to7=0.1*(28/(3π))≈0.297, correct.So, total≈5605.6 +1787.14 +2.1 +0.297≈7395.137But earlier, when I computed 5617.7333/π≈1788.5, and added to 5607.7, got≈7396.2The slight discrepancy is due to rounding.So, the total is approximately 7395.14 calories.But let me compute it more accurately.Compute 5616.8/π:π≈3.14159265365616.8 /3.1415926536≈Compute 3.1415926536*1787=5614.0So, 5616.8 -5614.0=2.8So, 2.8/3.1415926536≈0.891So, total≈1787 +0.891≈1787.891So, second integral≈1787.891Third integral=2.1Fourth integral≈0.297So, total≈5605.6 +1787.891 +2.1 +0.297≈5605.6 +1787.891=7393.491 7393.491 +2.1=7395.591 7395.591 +0.297≈7395.888So,≈7395.89 calories.Therefore, the total caloric expenditure for the week is approximately 7396 calories.But to be precise, let's compute it as:Total=5605.6 +1787.891 +2.1 +0.297=5605.6 +1787.891=7393.491 +2.1=7395.591 +0.297=7395.888≈7395.89So,≈7395.89 calories.But since the problem might expect an exact form, let me see:Total=5605.6 + (401.2*14)/π +2.1 + (0.1*28)/(3π)=5605.6 +5616.8/π +2.1 +2.8/(3π)=5607.7 + (5616.8 +2.8/3)/π=5607.7 + (5616.8 +0.9333)/π=5607.7 +5617.7333/πSo, if we leave it in terms of π, it's 5607.7 +5617.7333/πBut probably, the answer is expected to be a numerical value, so≈7396 calories.But let me check if I made a mistake in the first integral.Wait, the first integral was ∫800.8 dt from0 to7=800.8*7=5605.6, correct.Yes.So, the total is≈7396 calories.Therefore, the answers are:1. The range of H is approximately 0.5 to 0.75 hours.2. The total caloric expenditure for the week is approximately 7396 calories.But let me check if I can express the first answer more precisely.Earlier, we found that H must satisfy 200 ≤400H +0.1H³ ≤300We found that H≈0.5 and H≈0.75 are the bounds.But perhaps we can express the exact solutions.Let me try to solve 400H +0.1H³=2000.1H³ +400H -200=0Multiply by10: H³ +4000H -2000=0This is a cubic equation: H³ +4000H -2000=0We can try to find rational roots using Rational Root Theorem. Possible roots are factors of 2000 over factors of 1, so ±1, ±2, ±4, etc.Testing H=1:1 +4000 -2000=2001≠0 H=2:8 +8000 -2000=6008≠0 H=0.5:0.125 +2000 -2000=0.125≠0 H=0.4:0.064 +1600 -2000= -399.936≠0 H=0.5 is close, but not exact.Similarly, for H=0.5, the value is≈200.0125, which is just above 200.So, the exact solution is H≈0.5.Similarly, for the upper bound, solving 400H +0.1H³=3000.1H³ +400H -300=0 Multiply by10: H³ +4000H -3000=0Again, trying rational roots:H=1:1 +4000 -3000=1001≠0 H=2:8 +8000 -3000=5008≠0 H=0.75:0.421875 +3000 -3000=0.421875≠0 H=0.75 is close, but not exact.So, the exact solutions are H≈0.5 and H≈0.75.Therefore, the range of H is approximately 0.5 to 0.75 hours.So, summarizing:1. The range of H is approximately 0.5 to 0.75 hours.2. The total caloric expenditure for the week is approximately 7396 calories.</think>