🧠:<think>Okay, so I have this problem about a systems analyst integrating a new hardware-software system for a data center. There are two parts to the problem. Let me tackle them one by one.Starting with the first part: The data center has n servers, each can handle up to C concurrent connections. The load balancer distributes incoming requests which follow a Poisson distribution with rate λ per second. I need to find the conditions under which the probability of any server exceeding its capacity C is less than ε.Hmm, Poisson distribution is involved, so I think this relates to queueing theory. Each server can be modeled as a queue, right? So maybe each server is an M/M/1 queue or something similar. Wait, but the load balancing complicates things because requests are distributed across multiple servers.I remember that when you have multiple servers, the system can be modeled as an M/M/n queue. In such a system, the probability that a server is busy can be calculated, and from there, we can find the probability that a server exceeds its capacity.But in this case, each server has a maximum capacity C. So, actually, each server is an M/M/1/C queue. That is, each server can handle up to C connections, and any additional requests would be blocked or queued elsewhere. But in a load-balanced system, the load balancer would distribute the requests to different servers, so maybe each server doesn't see the full load.Wait, so the arrival rate per server would be λ/n, assuming the load is perfectly balanced. So each server has an arrival rate of λ/n and a service rate μ. But the problem doesn't specify the service rate, so maybe we can assume that each server can handle requests at a rate μ, and the utilization per server is λ/(nμ). But since each server can handle up to C connections, we need to ensure that the probability that the number of connections on any server exceeds C is less than ε. So, for each server, we can model it as a birth-death process with maximum capacity C.In an M/M/1/C queue, the probability that the server is in state k (i.e., has k connections) is given by:P(k) = (ρ^k / k!) * (1 - ρ) / (1 - ρ^{C+1} / (C+1)!))Wait, no, that's for M/M/1/C. Let me recall the formula correctly.In an M/M/1/C queue, the probability of being in state k is:P(k) = (ρ^k / k!) * (1 - ρ) / (1 - ρ^{C+1} / (C+1)!)) for k = 0, 1, ..., C.Wait, actually, no. The general formula for M/M/1/C is:P(k) = (ρ^k / k!) * (1 - ρ) / (1 - ρ^{C+1} / (C+1)!)) for k = 0, 1, ..., C.But I think that's not quite right. Let me think again.Actually, for an M/M/1/C queue, the stationary probabilities are:P(k) = (ρ^k / k!) * (1 - ρ) / (1 - ρ^{C+1} / (C+1)!)) for k = 0, 1, ..., C.Wait, no, that seems too complicated. Maybe it's simpler.In an M/M/1/C queue, the probability of being in state k is:P(k) = (ρ^k / k!) * (1 - ρ) / (1 - ρ^{C+1} / (C+1)!)).But actually, I think the denominator is the sum from k=0 to C of (ρ^k / k!). So, it's:P(k) = (ρ^k / k!) / (sum_{m=0}^C (ρ^m / m!)) )Yes, that makes sense. So, the stationary distribution is P(k) = (ρ^k / k!) / (sum_{m=0}^C (ρ^m / m!)) for k = 0, 1, ..., C.So, the probability that a server exceeds its capacity C is the probability that the number of connections is greater than C. But in this case, since the server can only handle up to C, the probability of exceeding is zero? Wait, no, because the load balancer is distributing the requests. So, actually, each server can have at most C connections, so the probability of exceeding is zero if the system is designed properly.Wait, maybe I'm misunderstanding. If the load balancer distributes the requests, then each server's load is λ/n. So, the utilization per server is ρ = (λ/n) / μ. So, as long as ρ < 1, the server can handle the load without growing indefinitely.But the question is about the probability that any server exceeds its maximum capacity C. So, in the M/M/1/C model, the probability that the server is in state C+1 is zero because it can't go beyond C. So, the probability of exceeding is zero. That doesn't make sense.Wait, perhaps the model is different. Maybe each server can have more than C connections, but we want the probability that it exceeds C to be less than ε. So, in that case, we can model each server as an M/M/1 queue without capacity limit, but calculate the probability that the number of customers exceeds C.In that case, for an M/M/1 queue, the stationary probability of having k customers is P(k) = (1 - ρ) ρ^k, where ρ = λ/μ.So, the probability that the number of customers exceeds C is sum_{k=C+1}^infty P(k) = sum_{k=C+1}^infty (1 - ρ) ρ^k = (1 - ρ) ρ^{C+1} / (1 - ρ) ) = ρ^{C+1}.Wait, that can't be. Let me compute it correctly.sum_{k=C+1}^infty (1 - ρ) ρ^k = (1 - ρ) ρ^{C+1} / (1 - ρ) ) = ρ^{C+1}.Wait, no, that's not correct. The sum from k=C+1 to infinity of ρ^k is ρ^{C+1} / (1 - ρ). So, multiplying by (1 - ρ), we get (1 - ρ) * ρ^{C+1} / (1 - ρ) ) = ρ^{C+1}.So, the probability that the number of customers exceeds C is ρ^{C+1}.Therefore, we want ρ^{C+1} < ε.But ρ is the utilization per server, which is (λ/n)/μ = λ/(nμ).So, substituting, we have (λ/(nμ))^{C+1} < ε.Therefore, the condition is λ/(nμ) < ε^{1/(C+1)}.So, n > λ/(μ ε^{1/(C+1)}).Therefore, the number of servers n must be greater than λ divided by μ times ε to the power of 1/(C+1).Wait, let me double-check.If each server has utilization ρ = λ/(nμ), then the probability that a server has more than C customers is ρ^{C+1}.So, setting ρ^{C+1} < ε, we get ρ < ε^{1/(C+1)}.Thus, λ/(nμ) < ε^{1/(C+1)}.Therefore, n > λ/(μ ε^{1/(C+1)}).So, that's the condition.But wait, is this correct? Because in reality, the load balancer distributes the requests, so each server's arrival rate is λ/n, and the service rate is μ, so ρ = (λ/n)/μ = λ/(nμ).Yes, that seems right.So, the condition is n > λ/(μ ε^{1/(C+1)}).Therefore, the number of servers must be greater than λ divided by μ times ε to the power of 1/(C+1).Okay, that seems reasonable.Now, moving on to the second part.We have a Markov Chain with transition matrix P:P = [ [0.1, 0.4, 0.5], [0.3, 0.5, 0.2], [0.4, 0.1, 0.5] ]We need to find the steady-state distribution vector π.The steady-state distribution π is a row vector such that π = π P, and the sum of the components of π is 1.So, let's denote π = [π1, π2, π3].Then, we have the following equations:π1 = 0.1 π1 + 0.3 π2 + 0.4 π3π2 = 0.4 π1 + 0.5 π2 + 0.1 π3π3 = 0.5 π1 + 0.2 π2 + 0.5 π3And π1 + π2 + π3 = 1.So, let's write these equations:1) π1 = 0.1 π1 + 0.3 π2 + 0.4 π32) π2 = 0.4 π1 + 0.5 π2 + 0.1 π33) π3 = 0.5 π1 + 0.2 π2 + 0.5 π34) π1 + π2 + π3 = 1Let me rearrange equations 1, 2, 3.From equation 1:π1 - 0.1 π1 - 0.3 π2 - 0.4 π3 = 00.9 π1 - 0.3 π2 - 0.4 π3 = 0 --> Equation 1'From equation 2:π2 - 0.4 π1 - 0.5 π2 - 0.1 π3 = 0-0.4 π1 + 0.5 π2 - 0.1 π3 = 0 --> Equation 2'From equation 3:π3 - 0.5 π1 - 0.2 π2 - 0.5 π3 = 0-0.5 π1 - 0.2 π2 + 0.5 π3 = 0 --> Equation 3'So, now we have three equations:1') 0.9 π1 - 0.3 π2 - 0.4 π3 = 02') -0.4 π1 + 0.5 π2 - 0.1 π3 = 03') -0.5 π1 - 0.2 π2 + 0.5 π3 = 0And equation 4: π1 + π2 + π3 = 1.So, we can solve this system.Let me write the equations in terms of variables:Equation 1': 0.9 π1 - 0.3 π2 - 0.4 π3 = 0Equation 2': -0.4 π1 + 0.5 π2 - 0.1 π3 = 0Equation 3': -0.5 π1 - 0.2 π2 + 0.5 π3 = 0Equation 4: π1 + π2 + π3 = 1I think it's easier to solve equations 1', 2', 3' and then use equation 4 to find the actual values.Let me write the coefficients matrix:Equation 1': 0.9, -0.3, -0.4Equation 2': -0.4, 0.5, -0.1Equation 3': -0.5, -0.2, 0.5Let me write this as a matrix:[ 0.9 -0.3 -0.4 ][ -0.4 0.5 -0.1 ][ -0.5 -0.2 0.5 ]We can solve this system using substitution or elimination.Alternatively, since it's a steady-state distribution, we can use the fact that the system is irreducible and aperiodic, so there's a unique stationary distribution.Alternatively, we can express π2 and π3 in terms of π1.Let me try to express π2 and π3 from equations 1' and 2'.From equation 1':0.9 π1 = 0.3 π2 + 0.4 π3So, π2 = (0.9 π1 - 0.4 π3)/0.3Similarly, from equation 2':-0.4 π1 + 0.5 π2 = 0.1 π3Let me substitute π2 from equation 1' into equation 2'.So, π2 = (0.9 π1 - 0.4 π3)/0.3Plugging into equation 2':-0.4 π1 + 0.5*(0.9 π1 - 0.4 π3)/0.3 = 0.1 π3Let me compute this step by step.First, compute 0.5*(0.9 π1 - 0.4 π3)/0.3:= (0.5 / 0.3)*(0.9 π1 - 0.4 π3)= (5/3)*(0.9 π1 - 0.4 π3)= (5/3)*0.9 π1 - (5/3)*0.4 π3= 1.5 π1 - (2/3) π3So, equation 2' becomes:-0.4 π1 + 1.5 π1 - (2/3) π3 = 0.1 π3Combine like terms:(-0.4 + 1.5) π1 + (-2/3 - 0.1) π3 = 01.1 π1 - (2/3 + 1/10) π3 = 0Convert 2/3 and 1/10 to common denominator:2/3 = 20/30, 1/10 = 3/30, so total is 23/30.So, 1.1 π1 - (23/30) π3 = 0Thus, 1.1 π1 = (23/30) π3So, π3 = (1.1 * 30 / 23) π1 = (33/23) π1 ≈ 1.4348 π1So, π3 = (33/23) π1Now, let's go back to equation 1':0.9 π1 = 0.3 π2 + 0.4 π3We have π3 = (33/23) π1, so:0.9 π1 = 0.3 π2 + 0.4*(33/23) π1Compute 0.4*(33/23):0.4 = 2/5, so 2/5 * 33/23 = 66/115 ≈ 0.5739So,0.9 π1 = 0.3 π2 + (66/115) π1Bring terms with π1 to the left:0.9 π1 - (66/115) π1 = 0.3 π2Compute 0.9 - 66/115:0.9 = 9/10 = 103.5/11566/115 is approximately 0.5739So, 103.5/115 - 66/115 = 37.5/115 = 75/230 = 15/46 ≈ 0.3261So,(15/46) π1 = 0.3 π2Thus, π2 = (15/46) π1 / 0.3 = (15/46) / (3/10) π1 = (15/46)*(10/3) π1 = (150/138) π1 = (25/23) π1 ≈ 1.087 π1So, π2 = (25/23) π1Now, we have π2 = (25/23) π1 and π3 = (33/23) π1Now, using equation 4: π1 + π2 + π3 = 1Substitute π2 and π3:π1 + (25/23) π1 + (33/23) π1 = 1Combine terms:π1*(1 + 25/23 + 33/23) = 1Compute the coefficients:1 = 23/2323/23 + 25/23 + 33/23 = (23 + 25 + 33)/23 = 81/23So,π1*(81/23) = 1Thus, π1 = 23/81 ≈ 0.28395Then, π2 = (25/23) π1 = (25/23)*(23/81) = 25/81 ≈ 0.3086π3 = (33/23) π1 = (33/23)*(23/81) = 33/81 = 11/27 ≈ 0.4074So, the steady-state distribution vector π is [23/81, 25/81, 33/81] which simplifies to [23/81, 25/81, 11/27].Wait, 33/81 is 11/27, yes.Let me check if these probabilities sum to 1:23/81 + 25/81 + 33/81 = (23 + 25 + 33)/81 = 81/81 = 1. Correct.Also, let's verify with equation 3':-0.5 π1 - 0.2 π2 + 0.5 π3 = 0Plugging in the values:-0.5*(23/81) - 0.2*(25/81) + 0.5*(33/81) =Compute each term:-0.5*(23/81) = -11.5/81-0.2*(25/81) = -5/810.5*(33/81) = 16.5/81Adding them up:-11.5/81 -5/81 +16.5/81 = (-11.5 -5 +16.5)/81 = 0/81 = 0. Correct.So, the steady-state distribution is π = [23/81, 25/81, 33/81] which can be simplified as [23/81, 25/81, 11/27].Alternatively, we can write all fractions with denominator 81:23/81, 25/81, 33/81.Yes, that's fine.So, to recap:π1 = 23/81π2 = 25/81π3 = 33/81 = 11/27So, that's the steady-state distribution.Final Answer1. The condition is ( n > frac{lambda}{mu epsilon^{1/(C+1)}} ), so the final answer is boxed{n > dfrac{lambda}{mu epsilon^{1/(C+1)}}}.2. The steady-state distribution vector ( pi ) is boxed{left( dfrac{23}{81}, dfrac{25}{81}, dfrac{11}{27} right)}.</think>

🧠:<think>Okay, so I have this problem about a golfer's strategy on a par-4 hole. There are two parts: the first shot is a drive with a given parabolic equation, and the second shot is an approach with parametric equations. I need to find the total horizontal distance for the drive and then the maximum height and total air time for the approach. Hmm, let's take it step by step.Starting with the first part: the drive's trajectory is given by ( y = -frac{1}{200}x^2 + x ). I remember that in projectile motion, the ball lands where it started vertically, so y would be 0. So, to find where the ball lands, I need to set y to 0 and solve for x. That should give me the horizontal distance.Let me write that equation down: ( 0 = -frac{1}{200}x^2 + x ). Hmm, I can factor this equation to solve for x. Let's see, factoring out an x gives me ( x(-frac{1}{200}x + 1) = 0 ). So, either x = 0 or ( -frac{1}{200}x + 1 = 0 ). Solving the second equation: ( -frac{1}{200}x + 1 = 0 ) leads to ( -frac{1}{200}x = -1 ), so multiplying both sides by -200 gives x = 200. So, the ball lands at x = 200 yards. That means the total horizontal distance is 200 yards. That seems straightforward.Wait, let me double-check. If I plug x = 200 back into the equation: ( y = -frac{1}{200}(200)^2 + 200 ). Calculating that: ( -frac{1}{200} * 40000 + 200 = -200 + 200 = 0 ). Yep, that checks out. So, the horizontal distance is indeed 200 yards.Moving on to the second part: the approach shot with parametric equations. The equations are:[x(t) = 60t y(t) = -16t^2 + 80t]They want the maximum height and the total time the ball is in the air. First, let's find the total time in the air. That's when the ball hits the ground again, so y(t) = 0. So, set ( -16t^2 + 80t = 0 ). Let's solve for t.Factor out a t: ( t(-16t + 80) = 0 ). So, t = 0 or ( -16t + 80 = 0 ). Solving the second equation: ( -16t = -80 ) leads to t = 5 seconds. So, the ball is in the air for 5 seconds. That makes sense because t = 0 is the start, and t = 5 is when it lands.Now, for the maximum height. In projectile motion, the maximum height occurs at the vertex of the parabola. Since the parametric equation for y is quadratic in t, the vertex will give the maximum height. The vertex occurs at t = -b/(2a) for a quadratic equation ( at^2 + bt + c ). Here, a = -16 and b = 80.So, t = -80/(2*(-16)) = -80/(-32) = 2.5 seconds. So, at t = 2.5 seconds, the ball reaches its maximum height. Let's plug t = 2.5 into the y(t) equation to find the height.Calculating y(2.5): ( y = -16*(2.5)^2 + 80*(2.5) ). First, ( (2.5)^2 = 6.25 ). Then, ( -16*6.25 = -100 ). And ( 80*2.5 = 200 ). So, y = -100 + 200 = 100 yards. So, the maximum height is 100 yards.Wait, that seems quite high for a golf shot. 100 yards is like a really high trajectory. Maybe I made a mistake? Let me check the calculations again.The parametric equation is ( y(t) = -16t^2 + 80t ). So, at t = 2.5, y = -16*(6.25) + 80*(2.5). That is indeed -100 + 200 = 100. Hmm, maybe in the problem, the units are different? Or perhaps it's correct because it's a wedge shot, which can have a high trajectory. I think the math checks out, so maybe 100 yards is correct.Alternatively, maybe the equation is in feet? But the problem says x and y are in yards for the first part, but for the second part, it's not specified. Wait, looking back: for the first part, x and y are in yards. For the second part, the parametric equations are given without units specified, but since x(t) is 60t and y(t) is -16t² +80t, and in the first part, x was in yards, maybe here it's also yards? But 16 is a familiar number in feet, since gravity is about 32 ft/s², so 16 is half of that. So, maybe the units here are in feet?Wait, the problem doesn't specify units for the second part. Hmm. So, perhaps I should assume that x(t) and y(t) are in yards as well? But 16 is a big number for yards. Wait, in the first part, the equation is in yards, but for the second part, the parametric equations are given with coefficients that might be in feet. Hmm, this is confusing.Wait, maybe the problem is consistent. Let me check: in the first part, the equation is in yards, so the units are yards. The second part, the parametric equations: x(t) = 60t. If x is in yards, then 60t would be yards per second. So, 60 yards per second is a very high speed, since 60 yards per second is like 216 km/h, which is way too fast for a golf ball. So, that can't be right.Alternatively, maybe x(t) is in feet. So, 60t would be in feet, meaning 60 feet per second. That's more reasonable, since 60 feet per second is about 40.9 mph, which is a typical golf ball speed. Similarly, y(t) is given as -16t² +80t. If y is in feet, then 16 is half of 32 ft/s², which is the acceleration due to gravity. So, that makes sense.Therefore, perhaps in the second part, the units are in feet, not yards. So, the maximum height is 100 feet, and the total time is 5 seconds. But the problem didn't specify, so that's a bit confusing. Wait, let me check the problem statement again.Looking back: "The key to their strategy is understanding the physics and geometry of their shots." Then, for the first shot, x and y are in yards. For the second shot, the parametric equations are given without units. Hmm. Maybe I should assume that the units are consistent with the first part, so yards. But as I saw, 60 yards per second is too fast. Alternatively, maybe the problem expects us to use the given equations regardless of units, so just take them as given.Wait, the problem says "x(t) = 60t" and "y(t) = -16t² +80t". So, if x(t) is in yards, then 60t is yards, so t must be in seconds, but 60 yards per second is too fast. Alternatively, if x(t) is in feet, then 60t is feet, so t is in seconds, which is more reasonable.But the problem didn't specify, so maybe we have to just go with the given equations and not worry about units? Or perhaps the problem assumes that both x and y are in yards for both parts? Hmm.Wait, in the first part, x and y are in yards. In the second part, the parametric equations are given without units, but since they are modeling the trajectory, it's likely that x and y are in yards as well. So, even though 60 yards per second seems high, maybe in the context of the problem, it's acceptable.Alternatively, perhaps the equations are in feet, and the answer should be in yards? Hmm, that complicates things. Wait, the problem says "the ball lands on the fairway" for the first shot, so it's in yards. For the second shot, it's approaching the green, so maybe it's also in yards. I'm a bit confused.Wait, maybe I should just proceed with the given equations, regardless of units, since the problem didn't specify. So, for the second part, the maximum height is 100 yards, and the total time is 5 seconds. But 100 yards is a huge height for a golf ball. The highest golf balls go maybe 50 yards high? So, 100 yards seems too much.Alternatively, maybe it's 100 feet, which would be about 30 meters, which is still very high, but more plausible. Wait, 100 feet is about 30 meters, which is still extremely high for a golf shot. Professional golfers can hit the ball up to 400 yards, but the maximum height is usually around 30-40 yards. So, 100 yards is way too high.Wait, maybe I made a mistake in the calculation. Let me recalculate y(2.5):y(t) = -16t² +80tAt t = 2.5:y = -16*(2.5)^2 +80*(2.5)First, 2.5 squared is 6.25.So, -16*6.25 = -10080*2.5 = 200So, y = -100 + 200 = 100.Hmm, that's correct. So, unless the units are different, the maximum height is 100. Maybe in the problem, it's 100 yards? Or maybe it's 100 feet? Since 16 is a familiar number in feet, perhaps the problem expects the answer in feet.But the problem didn't specify, so maybe I should just state it as 100 units, but that's not helpful. Alternatively, maybe the problem expects the answer in yards, so 100 yards, even though that seems high.Wait, let me think about the parametric equations. If x(t) = 60t, and y(t) = -16t² +80t, then if x is in yards, 60t is yards, so t is in seconds, and 60 yards per second is 216 km/h, which is too fast. If x is in feet, 60t is feet, so 60 feet per second is about 40.9 mph, which is more reasonable.Similarly, y(t) = -16t² +80t. If y is in feet, then 16 is half of 32 ft/s², which is correct. So, likely, the units are in feet. So, the maximum height is 100 feet, and the total time is 5 seconds.But the problem didn't specify, so maybe I should mention that? Or maybe just answer in the units given in the equations, which are consistent with feet.Wait, the problem says "where x and y are measured in yards" only for the first part. For the second part, it just says "where t is the time in seconds." So, maybe for the second part, x and y are in yards as well? Hmm, but then 60 yards per second is too fast.Alternatively, maybe the problem expects us to just use the equations as given, regardless of units, so the maximum height is 100, and the time is 5. But without units, it's unclear.Wait, maybe I should just proceed with the given equations, assuming that the units are consistent. So, for the second part, the maximum height is 100 yards, and the total time is 5 seconds. But I'm not sure if that's correct because of the unit inconsistency.Alternatively, maybe the problem expects the answer in yards, so 100 yards is the maximum height, even though it's high. I think I'll go with that, since the problem didn't specify units for the second part, but mentioned yards for the first part.Wait, no, the problem says "where t is the time in seconds" for the second part, but doesn't specify units for x(t) and y(t). So, maybe they are in yards? But as I saw, 60 yards per second is too fast. Hmm.Alternatively, maybe the problem is using a different gravity constant. In the first equation, the coefficient is -1/200, which is roughly -0.005, which is much less than the standard gravity. So, maybe in the first part, it's in yards, and in the second part, it's in feet? I'm getting confused.Wait, maybe I should just answer the questions as per the given equations, without worrying about the units, since the problem didn't specify. So, for the second part, the maximum height is 100, and the time is 5. But I need to specify units. Hmm.Wait, the problem says "x(t) = 60t" and "y(t) = -16t² +80t", where t is in seconds. So, x(t) is in yards? Or feet? Since the first part was in yards, maybe the second part is also in yards. So, 60 yards per second is too fast, but maybe in the context of the problem, it's acceptable.Alternatively, maybe the problem is using a different unit system. Wait, 16 is a familiar number in feet, as 16 is half of 32 ft/s². So, perhaps the problem is in feet, and the first part is in yards. So, for the second part, the maximum height is 100 feet, and the time is 5 seconds.But the problem didn't specify, so maybe I should just answer in the units given in the equations, which are consistent with feet. So, 100 feet and 5 seconds.Wait, but the first part was in yards, so maybe the second part is also in yards. So, 60 yards per second is too fast, but maybe in the problem's context, it's okay. So, the maximum height is 100 yards, and the time is 5 seconds.I think I need to make a decision here. Since the problem didn't specify units for the second part, but the first part was in yards, I'll assume that the second part is also in yards. So, the maximum height is 100 yards, and the time is 5 seconds. Even though 100 yards is very high, I'll go with that.Alternatively, maybe I should convert 100 feet to yards. Since 1 yard is 3 feet, 100 feet is about 33.33 yards. So, if the maximum height is 100 feet, that's approximately 33.33 yards. But the problem didn't specify, so I'm not sure.Wait, maybe I should just answer in the units given in the equations. Since the equations are given without units, but in the first part, it was yards, maybe the second part is also in yards. So, I'll proceed with that.So, summarizing:1. The drive travels 200 yards horizontally.2. The approach shot reaches a maximum height of 100 yards and is in the air for 5 seconds.But I'm still a bit unsure about the units for the second part. Maybe I should mention that the units are unclear, but based on the equations, the maximum height is 100 and the time is 5. Alternatively, if I consider the equations as feet, then the maximum height is 100 feet, which is about 30.48 meters, which is still very high for a golf shot.Wait, let me think about the initial velocity. For the approach shot, x(t) = 60t, so if x is in yards, then the horizontal speed is 60 yards per second, which is 60 * 0.9144 = 54.864 meters per second. That's extremely fast for a golf ball. The typical drive speed is around 100-140 mph, which is about 44.7 to 62.1 m/s. So, 54.864 m/s is within that range, but for an approach shot, which is usually a shorter club like a wedge, the speed is lower, maybe 50-80 mph, which is about 22.35 to 35.76 m/s. So, 54.864 m/s is a bit high for a wedge.Alternatively, if x(t) is in feet, then 60t is feet, so the horizontal speed is 60 feet per second, which is about 18.288 meters per second, which is more reasonable for a wedge shot.Similarly, the vertical motion: y(t) = -16t² +80t. If y is in feet, then the initial vertical speed is 80 ft/s, which is about 24.384 m/s. That's quite high for a wedge shot, but possible for a strong player.But again, the problem didn't specify, so I'm stuck. Maybe I should just answer in the units given, which are consistent with the equations, so 100 yards and 5 seconds. Or maybe 100 feet and 5 seconds. Hmm.Wait, another approach: in the first part, the equation is ( y = -frac{1}{200}x^2 + x ). The coefficient of x² is -1/200, which is approximately -0.005. If we compare that to the standard projectile motion equation, which is ( y = x tantheta - frac{g x^2}{2 v_0^2 cos^2theta} ), where g is acceleration due to gravity.In yards, g is approximately 32 ft/s², which is about 10.67 yards/s². So, the coefficient would be ( frac{g}{2 v_0^2 cos^2theta} ). If the coefficient is -1/200, then ( frac{10.67}{2 v_0^2 cos^2theta} = 1/200 ). Solving for ( v_0^2 cos^2theta ), we get ( v_0^2 cos^2theta = 10.67 * 200 / 2 = 1067 ). So, ( v_0 costheta = sqrt{1067} approx 32.67 ) yards per second. That's about 32.67 * 0.9144 ≈ 29.7 m/s, which is extremely high for a golf ball. So, maybe the units are different.Alternatively, if the first part is in feet, then g is 32 ft/s², so the coefficient would be ( frac{32}{2 v_0^2 cos^2theta} = 1/200 ). Then, ( v_0^2 cos^2theta = 32 * 200 / 2 = 3200 ). So, ( v_0 costheta = sqrt{3200} ≈ 56.57 ) ft/s, which is about 17.24 m/s, which is more reasonable for a drive.Wait, so if the first part is in feet, then the drive's horizontal distance would be 200 feet, which is about 61 meters, which is too short for a drive. Professional golfers can drive over 300 yards, which is about 274 meters. So, 200 feet is way too short.Wait, this is getting too complicated. Maybe I should just proceed with the given equations and answer the questions as they are, without worrying about the units, since the problem didn't specify for the second part.So, for the first part, the horizontal distance is 200 yards. For the second part, the maximum height is 100, and the time is 5. But I need to specify units. Since the first part was in yards, maybe the second part is also in yards, even though the numbers seem high. So, I'll go with that.Therefore, the answers are:1. 200 yards2. Maximum height: 100 yards, total time: 5 seconds.But I'm still unsure about the units for the second part. Maybe I should mention that the units are unclear, but based on the equations, the maximum height is 100 and the time is 5. Alternatively, if I consider the equations as feet, then the maximum height is 100 feet, which is about 30.48 meters, and the time is 5 seconds.Wait, maybe the problem expects the answer in yards for the first part and feet for the second part, but that's just a guess. Since the problem didn't specify, I think it's safer to assume that the units are consistent with the first part, so yards. So, I'll proceed with that.Final Answer1. The total horizontal distance is boxed{200} yards.2. The maximum height is boxed{100} yards and the total time in the air is boxed{5} seconds.</think>

🧠:<think>Okay, so I'm trying to help this screenwriter with their film script by adjusting the emotional engagement function. The function E(t) is piecewise sinusoidal, and they want to maximize the total emotional engagement over the entire timeline. Then, they also want to minimize the variance of E(t) while keeping the average at a target level. Hmm, that sounds a bit complex, but let's break it down step by step.First, for part 1, they want to maximize the integral of E(t) from t=0 to t=T3. The integral of E(t) over time would give the total emotional engagement, right? So, to maximize this, I need to figure out the best points T1 and T2 where the function changes its sinusoidal components.Looking at the function E(t), it's made up of three different sinusoidal functions, each active in different time intervals. Each of these has its own amplitude (A), frequency (B), phase shift (C), and vertical shift (D). The constants A, B, C, D are given for each segment, so I don't need to adjust those. Instead, I need to find the optimal T1 and T2.Wait, but the problem says the editor suggests adjusting the timing and intensity of emotional peaks and troughs. So, maybe the editor can change T1 and T2, but the screenwriter has already set A, B, C, D for each segment. Or is it that the editor can adjust all these parameters? Hmm, the problem statement says the screenwriter uses these constants to model different emotional arcs, so perhaps the editor is only adjusting T1 and T2, keeping A, B, C, D fixed? Or maybe the editor can adjust all parameters? Hmm, the problem says "the editor's insight to adjust the timing and intensity of emotional peaks and troughs within the script." So timing would relate to T1 and T2, and intensity might relate to A, B, C, D. Hmm, but in part 1, it's about maximizing the integral, so maybe just adjusting T1 and T2.Wait, the problem says "determine the values of T1 and T2 that maximize the integral of E(t) from t=0 to t=T3." So, only T1 and T2 are variables here, and the rest are constants. So, I need to find T1 and T2 such that the integral of E(t) over [0, T3] is maximized.Alright, so the integral is the sum of three integrals:Integral from 0 to T1 of E1(t) dt + Integral from T1 to T2 of E2(t) dt + Integral from T2 to T3 of E3(t) dt.Each Ei(t) is a sinusoidal function: Ai sin(Bi(t - Ci)) + Di.So, integrating each piece:Integral of Ai sin(Bi(t - Ci)) + Di dt = (-Ai/Bi) cos(Bi(t - Ci)) + Di*t + constant.So, evaluating each integral from their respective intervals:First integral: from 0 to T1:[-A1/B1 cos(B1(T1 - C1)) + D1*T1] - [-A1/B1 cos(B1(0 - C1)) + D1*0]= (-A1/B1 cos(B1(T1 - C1)) + D1*T1) - (-A1/B1 cos(-B1 C1))= (-A1/B1 cos(B1(T1 - C1)) + D1*T1) + A1/B1 cos(B1 C1)Similarly, the second integral from T1 to T2:[-A2/B2 cos(B2(T2 - C2)) + D2*T2] - [-A2/B2 cos(B2(T1 - C2)) + D2*T1]= (-A2/B2 cos(B2(T2 - C2)) + D2*T2) - (-A2/B2 cos(B2(T1 - C2)) + D2*T1)= (-A2/B2 cos(B2(T2 - C2)) + D2*T2) + A2/B2 cos(B2(T1 - C2)) - D2*T1Third integral from T2 to T3:[-A3/B3 cos(B3(T3 - C3)) + D3*T3] - [-A3/B3 cos(B3(T2 - C3)) + D3*T2]= (-A3/B3 cos(B3(T3 - C3)) + D3*T3) - (-A3/B3 cos(B3(T2 - C3)) + D3*T2)= (-A3/B3 cos(B3(T3 - C3)) + D3*T3) + A3/B3 cos(B3(T2 - C3)) - D3*T2So, the total integral is the sum of these three expressions.To maximize this, we need to take the derivative with respect to T1 and T2 and set them to zero.But wait, T1 and T2 are variables, so we can take partial derivatives.Let me denote the total integral as I(T1, T2):I = [(-A1/B1 cos(B1(T1 - C1)) + D1*T1) + A1/B1 cos(B1 C1)] + [(-A2/B2 cos(B2(T2 - C2)) + D2*T2) + A2/B2 cos(B2(T1 - C2)) - D2*T1] + [(-A3/B3 cos(B3(T3 - C3)) + D3*T3) + A3/B3 cos(B3(T2 - C3)) - D3*T2]Simplify this expression:I = (-A1/B1 cos(B1(T1 - C1)) + D1*T1 + A1/B1 cos(B1 C1)) + (-A2/B2 cos(B2(T2 - C2)) + D2*T2 + A2/B2 cos(B2(T1 - C2)) - D2*T1) + (-A3/B3 cos(B3(T3 - C3)) + D3*T3 + A3/B3 cos(B3(T2 - C3)) - D3*T2)Now, let's collect like terms:Terms with T1:D1*T1 - D2*T1Terms with T2:D2*T2 - D3*T2Constants:A1/B1 cos(B1 C1) - A3/B3 cos(B3(T3 - C3)) + D3*T3Terms with cosines:- A1/B1 cos(B1(T1 - C1)) - A2/B2 cos(B2(T2 - C2)) + A2/B2 cos(B2(T1 - C2)) + A3/B3 cos(B3(T2 - C3))So, I can write I as:I = (D1 - D2) T1 + (D2 - D3) T2 + [A1/B1 cos(B1 C1) - A3/B3 cos(B3(T3 - C3)) + D3*T3] + [- A1/B1 cos(B1(T1 - C1)) - A2/B2 cos(B2(T2 - C2)) + A2/B2 cos(B2(T1 - C2)) + A3/B3 cos(B3(T2 - C3))]Now, to find the maximum, take partial derivatives with respect to T1 and T2.Partial derivative of I with respect to T1:dI/dT1 = (D1 - D2) + [ derivative of the cosine terms with respect to T1 ]Looking at the cosine terms:- A1/B1 cos(B1(T1 - C1)) + A2/B2 cos(B2(T1 - C2))So, derivative:- A1/B1 * (-B1) sin(B1(T1 - C1)) + A2/B2 * (-B2) sin(B2(T1 - C2))Simplify:A1 sin(B1(T1 - C1)) - A2 sin(B2(T1 - C2))Similarly, partial derivative of I with respect to T2:dI/dT2 = (D2 - D3) + [ derivative of the cosine terms with respect to T2 ]Cosine terms:- A2/B2 cos(B2(T2 - C2)) + A3/B3 cos(B3(T2 - C3))Derivative:- A2/B2 * (-B2) sin(B2(T2 - C2)) + A3/B3 * (-B3) sin(B3(T2 - C3))Simplify:A2 sin(B2(T2 - C2)) - A3 sin(B3(T2 - C3))To maximize I, set both partial derivatives to zero:1. A1 sin(B1(T1 - C1)) - A2 sin(B2(T1 - C2)) + (D1 - D2) = 02. A2 sin(B2(T2 - C2)) - A3 sin(B3(T2 - C3)) + (D2 - D3) = 0So, these are the two equations we need to solve for T1 and T2.Hmm, these are transcendental equations, meaning they might not have analytical solutions and would require numerical methods to solve. But perhaps we can express the conditions in terms of the sine functions.Alternatively, if we consider that the maximum occurs when the derivatives are zero, which implies that the rate of change of the integral with respect to T1 and T2 is zero. So, the marginal contribution of extending T1 or T2 beyond that point doesn't increase the integral.But given that the equations involve sine functions, it's likely that we can't solve them algebraically. So, the screenwriter or editor would need to use numerical methods or optimization algorithms to find T1 and T2 that satisfy these conditions.Wait, but maybe there's a way to interpret this. The derivative with respect to T1 is the difference between the marginal contributions of the first and second segments. Similarly for T2, it's the difference between the second and third segments.So, setting these derivatives to zero would mean that the marginal contributions are balanced. That is, the rate at which the first segment contributes to the integral equals the rate at which the second segment does, and similarly for the second and third segments.But since the marginal contribution is given by the derivative of E(t) at T1 and T2, which for a sinusoidal function is the cosine term scaled by the amplitude and frequency.Wait, actually, the derivative of E(t) at T1 from the left is E1'(T1) = A1 B1 cos(B1(T1 - C1)), and from the right is E2'(T1) = A2 B2 cos(B2(T1 - C2)). For the integral to be maximized, the marginal contributions should be equal? Or perhaps the slopes should match? Hmm, but in this case, we're dealing with the derivative of the integral with respect to T1, which is the difference between the current segment's value and the next segment's value.Wait, no, actually, when you take the derivative of the integral with respect to T1, you're considering how much the integral changes if you move T1 a little bit. So, moving T1 to the right would add a little bit of the second segment and remove a little bit of the first segment. So, the derivative is E2(T1) - E1(T1). Similarly, the derivative with respect to T2 is E3(T2) - E2(T2).Wait, that makes sense. Because if you increase T1 by a small delta, the integral loses delta*E1(T1) and gains delta*E2(T1). So, the net change is delta*(E2(T1) - E1(T1)). To maximize the integral, this net change should be zero, so E2(T1) = E1(T1). Similarly, E3(T2) = E2(T2).Ah! So, that's a key insight. The optimal T1 and T2 are the points where the emotional engagement functions E1 and E2, and E2 and E3, respectively, intersect. That is, E1(T1) = E2(T1) and E2(T2) = E3(T2).So, to find T1, solve A1 sin(B1(T1 - C1)) + D1 = A2 sin(B2(T1 - C2)) + D2Similarly, for T2, solve A2 sin(B2(T2 - C2)) + D2 = A3 sin(B3(T2 - C3)) + D3These are equations in T1 and T2, respectively. Again, since they involve sine functions, they might not have closed-form solutions, so numerical methods would be necessary.But perhaps, if we make some assumptions about the parameters, we can find a solution. For example, if the functions are such that their peaks and troughs align in a certain way, or if the frequencies are the same, etc.Alternatively, if the screenwriter provides specific values for A1, B1, C1, D1, etc., we could plug them in and solve numerically. But since the problem doesn't give specific values, we can only provide the conditions that T1 and T2 must satisfy.So, summarizing part 1, the optimal T1 and T2 are the solutions to:A1 sin(B1(T1 - C1)) + D1 = A2 sin(B2(T1 - C2)) + D2andA2 sin(B2(T2 - C2)) + D2 = A3 sin(B3(T2 - C3)) + D3These equations must be solved numerically to find T1 and T2.Moving on to part 2, the editor wants to minimize the variance of E(t) over [0, T3], while keeping the average engagement equal to E_target.Variance is defined as E[E(t)^2] - (E[E(t)])^2. So, we need to minimize this quantity.Given that the average value of E(t) is E_target, which is a constant, we need to adjust the parameters A_i, B_i, C_i, D_i such that the variance is minimized.But wait, the problem says "determine the values of A_i, B_i, C_i, and D_i such that the variance is minimized while maintaining the condition that the average value of E(t) over [0, T3] is equal to a predefined target engagement level E_target."So, we have to minimize the variance subject to the average being E_target.First, let's express the average value:Average E = (1/T3) * Integral from 0 to T3 of E(t) dt = E_targetWe already have the integral from part 1, which is I(T1, T2). So, (1/T3)*I = E_target.So, we have a constraint: I = E_target * T3.And we need to minimize the variance, which is Var = (1/T3) Integral E(t)^2 dt - (E_target)^2.So, Var = (1/T3) Integral E(t)^2 dt - (E_target)^2.To minimize Var, we need to minimize Integral E(t)^2 dt, because (E_target)^2 is a constant.So, the problem reduces to minimizing Integral E(t)^2 dt subject to Integral E(t) dt = E_target * T3.This is a constrained optimization problem. We can use Lagrange multipliers.Let me denote the integral of E(t) as I and the integral of E(t)^2 as Q.We need to minimize Q subject to I = E_target * T3.So, set up the Lagrangian:L = Q - λ(I - E_target * T3)Take derivatives of L with respect to all variables (A1, B1, C1, D1, A2, B2, C2, D2, A3, B3, C3, D3, T1, T2) and set them to zero.But this seems extremely complex because we have many variables and the integrals involve sinusoidal functions.Alternatively, perhaps we can consider that the function E(t) is piecewise sinusoidal, and to minimize the variance, we need to make E(t) as constant as possible, because a constant function has zero variance. However, the average must be E_target, so the optimal E(t) would be E_target everywhere. But since E(t) is constrained to be piecewise sinusoidal, we need to make each segment as close to E_target as possible.Wait, but each segment is a sinusoidal function plus a vertical shift. So, if we set each sinusoidal component to zero, i.e., set A_i = 0, then E(t) would just be D_i, a constant. Then, to have the average E(t) = E_target, we can set D1 = D2 = D3 = E_target. Then, the variance would be zero, which is the minimum possible.But is that allowed? The problem says "the variance of the emotional engagement throughout the script should be minimized." So, if we can set each segment to a constant equal to E_target, that would minimize the variance to zero.But perhaps the screenwriter wants some variation in the emotional engagement, but the editor wants to minimize the variance. So, maybe the minimal variance is achieved when each segment is a constant function at E_target.But let's verify that.If E(t) is a constant function equal to E_target over the entire interval, then the variance is zero, which is indeed the minimum possible. So, to achieve this, set each A_i = 0 and each D_i = E_target. Then, the function E(t) is constant, and the variance is zero.But perhaps the screenwriter wants some variation, so maybe the editor can't set A_i to zero. But the problem doesn't specify any constraints on the parameters except that the average must be E_target. So, mathematically, the minimal variance is achieved when E(t) is constant, which requires A1 = A2 = A3 = 0 and D1 = D2 = D3 = E_target.But maybe the screenwriter wants the emotional arcs to have some variation, so perhaps the editor can't set A_i to zero. But the problem doesn't specify that, so perhaps the answer is to set all A_i = 0 and D_i = E_target.Alternatively, if the screenwriter insists on having sinusoidal components, then we need to find the parameters that make E(t) as close to E_target as possible, i.e., minimize the integral of (E(t) - E_target)^2 dt.But since the problem says "minimize the variance," which is equivalent to minimizing the integral of E(t)^2 dt given that the integral of E(t) dt is fixed.In that case, the minimal variance occurs when E(t) is as constant as possible, which again suggests setting A_i = 0 and D_i = E_target.But perhaps the screenwriter wants the emotional arcs to have some variation, so maybe the editor can't set A_i to zero. But the problem doesn't specify that, so perhaps the answer is to set all A_i = 0 and D_i = E_target.Alternatively, if we can't set A_i to zero, then we need to find the parameters that make E(t) as close to E_target as possible, which would involve setting the sinusoidal components to have zero mean and adjust the D_i to E_target.Wait, the average of a sinusoidal function over its period is zero, so if we set D_i = E_target, then the average of E(t) would be E_target, and the variance would be the average of (A_i sin(...))^2, which is (A_i^2)/2. So, to minimize the variance, we need to set A_i = 0, which again brings us back to the constant function.So, yes, the minimal variance is achieved when all A_i = 0 and D_i = E_target.But perhaps the screenwriter wants some variation, so maybe the editor can't set A_i to zero. But the problem doesn't specify that, so perhaps the answer is to set all A_i = 0 and D_i = E_target.Alternatively, if the screenwriter wants the emotional arcs to have some variation, the editor might need to adjust the parameters such that the sinusoidal components have minimal amplitude, i.e., A_i as small as possible, while keeping the average at E_target. But since the problem doesn't specify any constraints on the emotional arcs, the minimal variance is achieved when the function is constant.So, to answer part 2, the variance is minimized when each A_i = 0 and each D_i = E_target. Therefore, the function E(t) is constant at E_target throughout the script, resulting in zero variance.But wait, let's think again. The variance is defined as the average of (E(t) - mean)^2. If E(t) is constant, then variance is zero. If E(t) has any variation, the variance increases. So, yes, the minimal variance is zero, achieved when E(t) is constant.Therefore, the editor should set all A_i = 0 and all D_i = E_target.But perhaps the screenwriter wants some variation, so maybe the editor can't set A_i to zero. But the problem doesn't specify that, so perhaps the answer is to set all A_i = 0 and D_i = E_target.Alternatively, if the screenwriter insists on having sinusoidal components, then the editor needs to adjust the parameters such that the sinusoidal components have minimal impact on the variance. But since the variance is minimized when the function is constant, the only way is to set A_i = 0.So, in conclusion, for part 2, the minimal variance is achieved when all A_i = 0 and D_i = E_target.But let me double-check. If we set A_i = 0, then E(t) = D_i for each segment. To have the average E(t) = E_target, we need D1*(T1) + D2*(T2 - T1) + D3*(T3 - T2) = E_target*T3.If we set D1 = D2 = D3 = E_target, then the average is E_target, and the variance is zero.Yes, that makes sense.So, summarizing:1. To maximize the integral, set T1 and T2 such that E1(T1) = E2(T1) and E2(T2) = E3(T2).2. To minimize the variance, set A1 = A2 = A3 = 0 and D1 = D2 = D3 = E_target.But wait, in part 1, the editor is adjusting T1 and T2, but in part 2, the editor is adjusting A_i, B_i, C_i, D_i. So, in part 2, the editor can change all parameters, including T1 and T2, but the problem says "determine the values of A_i, B_i, C_i, and D_i such that the variance is minimized while maintaining the condition that the average value of E(t) over [0, T3] is equal to a predefined target engagement level E_target."So, T1 and T2 are part of the function, but the problem doesn't specify whether they can be adjusted or not. Wait, in part 1, the editor is adjusting T1 and T2, but in part 2, the problem says "determine the values of A_i, B_i, C_i, and D_i", so perhaps in part 2, T1 and T2 are fixed, and we need to adjust A_i, B_i, C_i, D_i.Wait, the problem says "the editor proposes that the variance of the emotional engagement throughout the script should be minimized. Given the same function E(t), calculate the variance of the function over the interval [0, T3] and determine the values of A_i, B_i, C_i, and D_i such that the variance is minimized while maintaining the condition that the average value of E(t) over [0, T3] is equal to a predefined target engagement level E_target."So, in part 2, the function E(t) is the same as in part 1, meaning T1 and T2 are fixed? Or are they variables? The problem doesn't specify, but since in part 1, the editor is adjusting T1 and T2, perhaps in part 2, T1 and T2 are fixed, and the editor is adjusting A_i, B_i, C_i, D_i.But the problem says "determine the values of A_i, B_i, C_i, and D_i", so perhaps T1 and T2 are fixed, and we need to adjust the other parameters.But if T1 and T2 are fixed, then the integral I is a function of A_i, B_i, C_i, D_i, and we need to minimize the variance, which is a function of E(t)^2, subject to the average being E_target.This is more complex because we have to consider the integrals over each segment.Let me denote:I = Integral E(t) dt = I1 + I2 + I3Where I1 = Integral from 0 to T1 of E1(t) dtI2 = Integral from T1 to T2 of E2(t) dtI3 = Integral from T2 to T3 of E3(t) dtSimilarly, Q = Integral E(t)^2 dt = Q1 + Q2 + Q3Where Qi = Integral of Ei(t)^2 dt over their respective intervals.We need to minimize Q subject to I = E_target * T3.So, set up the Lagrangian:L = Q1 + Q2 + Q3 - λ(I1 + I2 + I3 - E_target*T3)Take partial derivatives with respect to each parameter A_i, B_i, C_i, D_i and set them to zero.But this is going to be very involved because each integral Qi and Ii depends on multiple parameters.Alternatively, perhaps we can consider each segment separately and find the optimal parameters for each segment that minimize the contribution to Q while satisfying the contribution to I.But since the segments are connected at T1 and T2, which are fixed, the parameters for each segment can be optimized independently, except for the fact that the overall average must be E_target.Wait, but each segment's integral contributes to the total I, so we have to ensure that the sum of I1 + I2 + I3 = E_target*T3.So, perhaps we can treat each segment's contribution to I and Q separately, and then find the parameters that minimize Q given the total I.But this is getting quite complex. Maybe a better approach is to consider that for each segment, the function Ei(t) = Ai sin(Bi(t - Ci)) + Di.To minimize the variance, we need to minimize the integral of Ei(t)^2 dt over each interval, subject to the integral of Ei(t) dt over each interval contributing to the total I.But since the total I is fixed, perhaps we can use the method of Lagrange multipliers for each segment.For each segment, the contribution to I is Ii = Integral Ei(t) dt = (-Ai/Bi cos(Bi(Ti - Ci)) + Di*Ti) - [(-Ai/Bi cos(Bi(0 - Ci)) + Di*0)] for the first segment, and similarly for others.But this is getting too involved. Maybe a simpler approach is to realize that for a given segment, the function Ei(t) = Ai sin(Bi(t - Ci)) + Di.The average of Ei(t) over its interval is (Ii)/(length of interval). To minimize the variance, we need to minimize the integral of (Ei(t) - average)^2 dt.But since the overall average is fixed, perhaps each segment's average should be equal to E_target. Because if each segment's average is E_target, then the overall average is E_target, and the variance is the sum of the variances of each segment.But wait, no, because the segments have different lengths. So, the overall average is a weighted average of the segment averages, weighted by their lengths.So, if we denote the length of the first segment as L1 = T1, second as L2 = T2 - T1, third as L3 = T3 - T2.Then, the overall average is (L1*avg1 + L2*avg2 + L3*avg3)/(L1 + L2 + L3) = E_target.To minimize the total variance, we need to minimize the sum of the variances of each segment, which is the sum of Integral (Ei(t) - avg_i)^2 dt over each interval.But since the overall average is E_target, which is a weighted average of the segment averages, perhaps the minimal total variance occurs when each segment's average is equal to E_target, and the segments are as flat as possible.Wait, but if each segment's average is E_target, then the overall average is E_target, and the variance is the sum of the variances of each segment.To minimize the total variance, each segment should have minimal variance, which occurs when each segment is a constant function equal to E_target. So, again, set Ai = 0 and Di = E_target for each segment.But if the screenwriter wants some variation, maybe the editor can't set Ai = 0. But the problem doesn't specify that, so perhaps the minimal variance is achieved when each segment is constant at E_target.Therefore, the answer for part 2 is to set all Ai = 0 and Di = E_target.But let me think again. If we set Ai = 0, then E(t) = Di for each segment. To have the overall average E_target, we need (L1*D1 + L2*D2 + L3*D3)/(L1 + L2 + L3) = E_target.If we set D1 = D2 = D3 = E_target, then the overall average is E_target, and the variance is zero.Yes, that makes sense.So, in conclusion:1. To maximize the integral, set T1 and T2 such that E1(T1) = E2(T1) and E2(T2) = E3(T2).2. To minimize the variance, set A1 = A2 = A3 = 0 and D1 = D2 = D3 = E_target.But wait, in part 1, the editor is adjusting T1 and T2, but in part 2, the editor is adjusting A_i, B_i, C_i, D_i. So, in part 2, T1 and T2 are fixed, and we need to adjust the other parameters.But if T1 and T2 are fixed, and we set Ai = 0 and Di = E_target, then the function is constant at E_target, regardless of T1 and T2. So, the variance is zero.Yes, that's correct.So, the final answers are:1. T1 and T2 are the solutions to E1(T1) = E2(T1) and E2(T2) = E3(T2).2. Set A1 = A2 = A3 = 0 and D1 = D2 = D3 = E_target.But the problem says "determine the values of A_i, B_i, C_i, and D_i", so we can set B_i and C_i to any values, but since Ai = 0, the phase and frequency don't matter. So, we can set B_i and C_i to any values, but to minimize the variance, we set Ai = 0 and Di = E_target.Therefore, the final answer is:For part 1, T1 and T2 are the solutions to the equations where the emotional engagement functions intersect at those points.For part 2, set all A_i to zero and all D_i to E_target.But let me write this more formally.For part 1:The optimal T1 and T2 are found by solving:A1 sin(B1(T1 - C1)) + D1 = A2 sin(B2(T1 - C2)) + D2andA2 sin(B2(T2 - C2)) + D2 = A3 sin(B3(T2 - C3)) + D3These equations must be solved numerically.For part 2:To minimize the variance, set A1 = A2 = A3 = 0 and D1 = D2 = D3 = E_target. This makes E(t) constant at E_target, resulting in zero variance.So, summarizing:1. T1 and T2 are the solutions to the above equations.2. A_i = 0, D_i = E_target.But let me check if setting A_i = 0 and D_i = E_target satisfies the average condition. Yes, because each segment's average is E_target, so the overall average is E_target.Yes, that's correct.So, the final answers are:1. T1 and T2 are the solutions to E1(T1) = E2(T1) and E2(T2) = E3(T2).2. A1 = A2 = A3 = 0 and D1 = D2 = D3 = E_target.But in the problem statement, part 2 says "determine the values of A_i, B_i, C_i, and D_i", so we need to specify all of them. Since Ai = 0, the values of Bi and Ci don't affect the function, so they can be arbitrary. But to fully specify, perhaps we can set Bi and Ci to any values, but since Ai = 0, the function is just Di, so Bi and Ci are irrelevant.Therefore, the answer for part 2 is:Set A1 = A2 = A3 = 0, and D1 = D2 = D3 = E_target. The values of B_i and C_i can be arbitrary since they don't affect the function when A_i = 0.But perhaps the problem expects us to set B_i and C_i to specific values, but since Ai = 0, they don't matter. So, we can just state that A_i = 0 and D_i = E_target, and B_i and C_i can be any constants.Therefore, the final answers are:1. T1 and T2 are the solutions to the equations where E1(T1) = E2(T1) and E2(T2) = E3(T2).2. A1 = A2 = A3 = 0 and D1 = D2 = D3 = E_target.</think>

🧠:<think>Okay, so I have this math problem about Mrs. Gertrude and her cookie distribution. Let me try to figure it out step by step. First, problem 1 says that Mrs. Gertrude distributes cookies in a geometric sequence. She starts with 128 cookies on the first day and gives away half the number of cookies each subsequent day. I need to find the total number of cookies she gives away after 'n' days and provide a general formula for that.Hmm, geometric sequence. I remember that a geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio. In this case, the first term is 128 cookies, and each day she gives away half the number of cookies from the previous day. So, the common ratio 'r' is 1/2.The formula for the sum of the first 'n' terms of a geometric series is S_n = a1 * (1 - r^n) / (1 - r), where a1 is the first term, r is the common ratio, and n is the number of terms. Let me plug in the values. a1 is 128, r is 1/2. So, S_n = 128 * (1 - (1/2)^n) / (1 - 1/2). Simplifying the denominator, 1 - 1/2 is 1/2. So, S_n = 128 * (1 - (1/2)^n) / (1/2). Dividing by 1/2 is the same as multiplying by 2, so S_n = 128 * 2 * (1 - (1/2)^n). Calculating that, 128 * 2 is 256. So, S_n = 256 * (1 - (1/2)^n). That should be the total number of cookies given away after 'n' days. Let me double-check my steps. Starting with 128, each day halves, so the sequence is 128, 64, 32, 16, etc. The sum formula for a geometric series is correct, and plugging in the values seems right. Yeah, I think that's the correct formula.Moving on to problem 2. Mrs. Gertrude lives on a street where the house numbers are in an arithmetic sequence. The first house number is 101, and the common difference is 3. I need to find the house number of the 50th house. Additionally, if she visits every 10th house starting from her own, determine the house numbers she will visit up to the 50th house.Alright, arithmetic sequence. The formula for the nth term of an arithmetic sequence is a_n = a1 + (n - 1)d, where a1 is the first term, d is the common difference, and n is the term number. So, for the 50th house, a50 = 101 + (50 - 1)*3. Let me compute that. 50 - 1 is 49, multiplied by 3 is 147. Adding that to 101 gives 101 + 147 = 248. So, the 50th house number is 248. That seems straightforward.Now, the second part: Mrs. Gertrude visits every 10th house starting from her own. I need to figure out the house numbers she visits up to the 50th house. Wait, starting from her own house. So, if her house is the first house, number 101, then she visits every 10th house. So, the sequence would be 101, 101 + 10*3, 101 + 20*3, etc., but wait, no. Wait, the common difference is 3, so each subsequent house is 3 more than the previous. But she's visiting every 10th house, meaning she skips 9 houses each time.Wait, maybe I need to think of it differently. If she starts at her own house, which is the first house, number 101, then the next house she visits would be the 11th house, then the 21st, 31st, 41st, and so on, up to the 50th house. So, the sequence of house numbers she visits would be the 1st, 11th, 21st, 31st, 41st, and 51st, but since the 51st is beyond 50, we stop at the 50th. Wait, but 50 is the last house, so maybe she only goes up to the 50th. Let me clarify.Wait, the problem says "up to the 50th house." So, she starts at her own house (1st), then the 11th, 21st, 31st, 41st, and then the next would be 51st, which is beyond 50, so she stops at 50. But wait, is 50 included? Let me see.Alternatively, maybe she starts at her own house, which is the first house, and then every 10th house after that. So, the sequence would be 1st, 11th, 21st, 31st, 41st, and 51st. But since the 51st is beyond 50, she doesn't visit it. So, the houses she visits are 1st, 11th, 21st, 31st, 41st. But wait, the problem says "starting from her own," which is the first house, and "visits every 10th house." So, the first house is 101, then the 11th house, which is 101 + 10*3 = 101 + 30 = 131? Wait, no, because the common difference is 3, so each house is 3 more than the previous. So, the nth house is 101 + (n - 1)*3. So, the 1st house is 101, the 2nd is 104, the 3rd is 107, and so on. So, the 11th house would be 101 + (11 - 1)*3 = 101 + 30 = 131. Similarly, the 21st house is 101 + 20*3 = 101 + 60 = 161. The 31st is 101 + 30*3 = 101 + 90 = 191. The 41st is 101 + 40*3 = 101 + 120 = 221. The 51st would be 101 + 50*3 = 101 + 150 = 251, but that's beyond the 50th house, which is 248. So, she only visits up to the 41st house, which is 221. Wait, but the 50th house is 248, so maybe she can visit the 50th house if it's a multiple of 10? Let me check. The 50th house is 248, which is the 50th term. Is 50 a multiple of 10? Yes, 50 is 5*10, so she would visit the 50th house as well. So, the sequence would be 1st, 11th, 21st, 31st, 41st, and 50th. Wait, 50 is not 51, so 50 is the 50th term. Let me compute the 50th term, which we already did as 248. So, she visits the 1st, 11th, 21st, 31st, 41st, and 50th houses. Wait, but 50 is not a multiple of 10, it's 5*10, but in terms of the sequence, the 50th house is the 50th term, so if she starts at 1, then every 10th term, so 1, 11, 21, 31, 41, 51. But 51 is beyond 50, so she stops at 50. But 50 is not a multiple of 10 in terms of the term number, but it's the 50th term. Hmm, maybe I need to clarify.Alternatively, perhaps she visits every 10th house in terms of house numbers, not term numbers. But the problem says "visits every 10th house starting from her own." So, starting from her own house (1st), then the next would be the 11th house, then 21st, etc., regardless of the house numbers. So, in terms of term numbers, she visits term 1, term 11, term 21, term 31, term 41, term 51, but since term 51 is beyond 50, she stops at term 50. But term 50 is the 50th house, which is 248. So, does she visit the 50th house? Wait, if she starts at term 1, then every 10th term after that. So, term 1, term 11, term 21, term 31, term 41, term 51. But term 51 is beyond 50, so she doesn't visit it. Therefore, the houses she visits are term 1, 11, 21, 31, 41. So, their house numbers are:Term 1: 101Term 11: 101 + (11 - 1)*3 = 101 + 30 = 131Term 21: 101 + (21 - 1)*3 = 101 + 60 = 161Term 31: 101 + (31 - 1)*3 = 101 + 90 = 191Term 41: 101 + (41 - 1)*3 = 101 + 120 = 221Term 51 would be 101 + 150 = 251, which is beyond 50, so she doesn't visit it. Therefore, the house numbers she visits are 101, 131, 161, 191, 221.Wait, but the 50th house is 248, which is term 50. So, if she is visiting every 10th house starting from her own, does she visit the 50th house? Let's see, term 50 is 50, which is 5*10, so yes, it's a multiple of 10. So, she would visit term 50 as well. Therefore, the house numbers would be 101, 131, 161, 191, 221, and 248.Wait, but term 50 is 248, which is 101 + (50 - 1)*3 = 101 + 147 = 248. So, yes, she would visit term 50 as well, since 50 is a multiple of 10. Therefore, the house numbers she visits are 101, 131, 161, 191, 221, and 248.Wait, but let me think again. If she starts at term 1, then every 10th term after that. So, term 1, then term 1 + 10 = term 11, then term 11 + 10 = term 21, and so on. So, term 1, 11, 21, 31, 41, 51. But term 51 is beyond 50, so she stops at term 50. But term 50 is not a multiple of 10 in terms of the term number, because 50 divided by 10 is 5, but term 50 is the 50th term, which is 5*10, so yes, it's a multiple of 10. Therefore, she does visit term 50. So, the house numbers are 101, 131, 161, 191, 221, and 248.Wait, but let me check: term 1: 101term 11: 101 + 10*3 = 131term 21: 101 + 20*3 = 161term 31: 101 + 30*3 = 191term 41: 101 + 40*3 = 221term 51: 101 + 50*3 = 251 (beyond 50)But term 50 is 101 + 49*3 = 248, which is not a multiple of 10 in terms of term number, but it's the 50th term, which is 5*10, so it's a multiple of 10. Therefore, she visits term 50 as well. So, the house numbers are 101, 131, 161, 191, 221, and 248.Wait, but term 50 is 248, which is the 50th term, so she does visit it because 50 is a multiple of 10. So, the sequence is 101, 131, 161, 191, 221, 248.Alternatively, maybe she doesn't visit term 50 because she's only supposed to visit every 10th house starting from her own, meaning term 1, 11, 21, 31, 41, 51, etc., and since 51 is beyond 50, she stops at term 41. So, the house numbers would be 101, 131, 161, 191, 221.I think the confusion arises from whether term 50 is considered a multiple of 10. Since 50 is 5*10, it is a multiple of 10, so she would visit it. Therefore, the house numbers she visits are 101, 131, 161, 191, 221, and 248.Wait, but let me think again. If she starts at term 1, then the next house she visits is term 1 + 10 = term 11, then term 21, term 31, term 41, term 51. Since term 51 is beyond 50, she doesn't visit it. Therefore, the last house she visits is term 41, which is 221. So, the house numbers are 101, 131, 161, 191, 221.But wait, term 50 is 248, which is the 50th term, and 50 is a multiple of 10, so she should visit it as well. Therefore, she visits term 50 as well. So, the house numbers are 101, 131, 161, 191, 221, and 248.I think the correct approach is that she visits every 10th term starting from term 1, so term 1, 11, 21, 31, 41, 51, etc. Since 51 is beyond 50, she stops at term 50 if it's a multiple of 10. Since 50 is a multiple of 10, she visits term 50. Therefore, the house numbers are 101, 131, 161, 191, 221, and 248.Wait, but term 50 is 248, which is 101 + (50 - 1)*3 = 101 + 147 = 248. So, yes, she visits term 50 as well. Therefore, the house numbers are 101, 131, 161, 191, 221, and 248.I think that's the correct answer. So, summarizing:Problem 1: The total number of cookies given away after 'n' days is 256*(1 - (1/2)^n).Problem 2: The 50th house number is 248. The house numbers she visits are 101, 131, 161, 191, 221, and 248.Wait, but let me double-check the arithmetic for the house numbers she visits. Term 1: 101Term 11: 101 + (11-1)*3 = 101 + 30 = 131Term 21: 101 + (21-1)*3 = 101 + 60 = 161Term 31: 101 + (31-1)*3 = 101 + 90 = 191Term 41: 101 + (41-1)*3 = 101 + 120 = 221Term 50: 101 + (50-1)*3 = 101 + 147 = 248Yes, that's correct. So, she visits these six houses: 101, 131, 161, 191, 221, 248.I think that's it. I hope I didn't make any mistakes.</think>