🧠:<think>Alright, so I have two sub-problems to solve here. Let me tackle them one by one.Starting with Sub-problem 1: It's about finding an explicit formula for a recursive sequence. The sequence is defined by (a_{n+2} = 3a_{n+1} - 2a_n) with initial terms (a_0 = 2) and (a_1 = 5). Hmm, okay, this looks like a linear recurrence relation. I remember that for such recursions, we can find a characteristic equation and solve for its roots to get the general solution.So, the characteristic equation for (a_{n+2} = 3a_{n+1} - 2a_n) would be (r^2 - 3r + 2 = 0). Let me solve that quadratic equation. The discriminant is (9 - 8 = 1), so the roots are (r = frac{3 pm 1}{2}). That gives (r = 2) and (r = 1). Since we have two distinct real roots, the general solution should be (a_n = A(2)^n + B(1)^n), where A and B are constants to be determined by the initial conditions.Now, let's plug in the initial terms to find A and B. For (n = 0): (a_0 = 2 = A(2)^0 + B(1)^0 = A + B). So, equation 1: (A + B = 2).For (n = 1): (a_1 = 5 = A(2)^1 + B(1)^1 = 2A + B). So, equation 2: (2A + B = 5).Now, subtract equation 1 from equation 2: (2A + B - (A + B) = 5 - 2), which simplifies to (A = 3).Substituting A back into equation 1: (3 + B = 2), so (B = -1).Therefore, the explicit formula should be (a_n = 3(2)^n - 1). Let me check with (n = 0): (3(1) - 1 = 2), which is correct. For (n = 1): (3(2) - 1 = 6 - 1 = 5), also correct. Let me test (n = 2): Using the recursion, (a_2 = 3a_1 - 2a_0 = 15 - 4 = 11). Using the formula: (3(4) - 1 = 12 - 1 = 11). Perfect, that matches. So I think this is correct.Moving on to Sub-problem 2: It's about the total area of an infinite series of nested equilateral triangles. Each triangle is inscribed in a circle, and the next triangle is formed by connecting the midpoints of the sides of the previous one. The side length of the largest triangle is (s). I need to find the total area.Okay, so first, let's recall the area of an equilateral triangle with side length (s). The formula is (frac{sqrt{3}}{4} s^2). So the area of the first triangle is (A_1 = frac{sqrt{3}}{4} s^2).Now, each subsequent triangle is formed by connecting the midpoints of the sides of the previous triangle. I think this process creates a smaller equilateral triangle inside the previous one. I need to find the side length of each subsequent triangle.When you connect the midpoints of an equilateral triangle, the new triangle is similar to the original one, scaled down by a factor. Let me figure out the scaling factor.In an equilateral triangle, connecting midpoints divides each side into two equal parts. So the new triangle's side length is half of the original? Wait, no. Wait, actually, when you connect midpoints, the new triangle is similar but scaled by a factor of 1/2? Hmm, let me think.Wait, no, connecting midpoints in a triangle actually creates four smaller triangles, each similar to the original, each with 1/4 the area. But in this case, the next triangle is formed by connecting midpoints, so the side length of the new triangle is half the original? Or is it something else?Wait, perhaps I should visualize it. If I have an equilateral triangle with side length (s), and I connect the midpoints of each side, the new triangle formed inside will have side length (s/2). Because each side of the new triangle is connecting midpoints, so each side is half the length of the original triangle's side.Wait, is that correct? Let me think about coordinates. Suppose the original triangle has vertices at (0,0), (s,0), and (s/2, (s√3)/2). The midpoints would be at (s/2, 0), (3s/4, (s√3)/4), and (s/4, (s√3)/4). Connecting these midpoints would form a smaller equilateral triangle. The distance between (s/2, 0) and (3s/4, (s√3)/4) is sqrt[(3s/4 - s/2)^2 + ((s√3)/4 - 0)^2] = sqrt[(s/4)^2 + (s√3/4)^2] = sqrt[(s²/16) + (3s²/16)] = sqrt[4s²/16] = sqrt[s²/4] = s/2. So yes, the side length of the new triangle is indeed s/2.So each subsequent triangle has half the side length of the previous one. Therefore, the side lengths form a geometric sequence: (s, s/2, s/4, s/8, ldots)Therefore, the areas would be: (A_1 = frac{sqrt{3}}{4} s^2), (A_2 = frac{sqrt{3}}{4} (s/2)^2 = frac{sqrt{3}}{4} s^2 /4 = A_1 /4), (A_3 = A_2 /4 = A_1 /16), and so on.So the areas form a geometric series where each term is 1/4 of the previous term. So the total area is the sum of this infinite geometric series.The formula for the sum of an infinite geometric series is (S = a_1 / (1 - r)), where (a_1) is the first term and (r) is the common ratio.Here, (a_1 = frac{sqrt{3}}{4} s^2) and (r = 1/4). So the total area (S = frac{sqrt{3}}{4} s^2 / (1 - 1/4) = frac{sqrt{3}}{4} s^2 / (3/4) = frac{sqrt{3}}{4} s^2 * (4/3) = frac{sqrt{3}}{3} s^2).Wait, but hold on. Is that correct? Because each time we connect midpoints, we're creating a new triangle, but are we adding the area of each new triangle to the total? Or is the total area just the sum of all these triangles?Wait, the problem says "the total area of the infinite series of triangles." So I think it's the sum of all the areas of each triangle in the series. So the first triangle has area (A_1), the second (A_2 = A_1 /4), the third (A_3 = A_2 /4 = A_1 /16), etc. So yes, the total area is (A_1 + A_2 + A_3 + ldots = A_1 (1 + 1/4 + 1/16 + 1/64 + ldots)).Which is indeed a geometric series with first term (A_1) and ratio (1/4). So the sum is (A_1 / (1 - 1/4) = ( sqrt{3}/4 s^2 ) / (3/4 ) = sqrt{3}/3 s^2).Wait, but let me confirm if the side length halves each time. Because when I connected midpoints, the side length of the new triangle is half, so the area is 1/4. So yes, each subsequent triangle is 1/4 the area of the previous one.Alternatively, another way to think about it is that each time, the area is scaled by (1/2)^2 = 1/4, since linear dimensions are scaled by 1/2.Therefore, the total area is indeed (sqrt{3}/3 s^2).But let me think again: is the total area just the sum of all these triangles? Because each triangle is nested inside the previous one, so the total area would be the area of the largest triangle plus the area of the next one, and so on. But wait, actually, each subsequent triangle is entirely inside the previous one, so the total area isn't just the sum because they overlap. Wait, hold on, that might be a mistake.Wait, no, the problem says "the total area of the infinite series of triangles." It doesn't specify whether it's the union or the sum. But in the context of a stained glass window, it's likely referring to the sum of all the areas, even if they overlap, because each triangle is a separate piece. So, in that case, the total area would indeed be the sum of all the areas, which is an infinite series.But wait, another thought: when you connect midpoints, each new triangle is actually 1/4 the area of the previous one, but it's placed inside. So the total area covered by all triangles would be the area of the largest triangle plus the area of the next one, etc., but since each subsequent triangle is entirely within the previous, the union's area is just the area of the largest triangle. But the problem says "the total area of the infinite series of triangles," which might mean the sum of their areas, regardless of overlap. So, in that case, it's the sum.But to be safe, let me think about the problem statement again: "the total area of the infinite series of triangles." It doesn't specify union or sum, but in mathematical terms, when talking about the total area of a series, it usually refers to the sum of their areas. So I think my initial approach is correct.Therefore, the total area is (sqrt{3}/3 s^2). Alternatively, that can be written as (frac{sqrt{3}}{3} s^2).Wait, but just to make sure, let me compute the first few terms:- First triangle: (A_1 = sqrt{3}/4 s^2 ≈ 0.4330 s^2)- Second triangle: (A_2 = sqrt{3}/4 (s/2)^2 = sqrt{3}/4 s^2 /4 = sqrt{3}/16 s^2 ≈ 0.10825 s^2)- Third triangle: (A_3 = sqrt{3}/4 (s/4)^2 = sqrt{3}/4 s^2 /16 = sqrt{3}/64 s^2 ≈ 0.02706 s^2)- And so on.So the total area would be approximately 0.4330 + 0.10825 + 0.02706 + ... which converges to (sqrt{3}/3 s^2 ≈ 0.577 s^2). That seems reasonable.Alternatively, if we think of it as the sum of an infinite geometric series with first term (A_1 = sqrt{3}/4 s^2) and ratio (1/4), then the sum is indeed (A_1 / (1 - 1/4) = (sqrt{3}/4 s^2) / (3/4) = sqrt{3}/3 s^2).So I think that's the answer.Final AnswerSub-problem 1: boxed{a_n = 3 cdot 2^n - 1}Sub-problem 2: boxed{dfrac{sqrt{3}}{3} s^2}</think>

🧠:<think>Okay, so I have this problem about a music photographer who's been taking photos for 20 years, and I need to figure out two things. First, the total number of photographs taken over those 20 years using a definite integral. Second, I need to find a constant ( k ) that scales the number of photos each year to the available wall space. Let me take this step by step.Starting with the first part: The photographer takes ( f(t) = 50 + 25t - 0.5t^2 ) photos each year, where ( t ) is the number of years since they started. I need to find the total number of photographs over 20 years. Hmm, so since ( f(t) ) gives the number of photos per year, integrating this function from 0 to 20 should give the total number of photos. That makes sense because integration sums up the area under the curve, which in this case would be the total photos over the 20-year span.So, the integral I need to compute is:[int_{0}^{20} (50 + 25t - 0.5t^2) , dt]Alright, let me break this down. I can integrate each term separately. The integral of 50 with respect to ( t ) is straightforward. The integral of 25t is also simple, and then the integral of -0.5t² should be manageable.Let me compute each integral term by term.First term: ( int 50 , dt )The integral of 50 is 50t. So, evaluated from 0 to 20, that would be 50*(20) - 50*(0) = 1000 - 0 = 1000.Second term: ( int 25t , dt )The integral of 25t is (25/2)t². So, evaluated from 0 to 20, that's (25/2)*(20)² - (25/2)*(0)². Let's compute that.First, (20)² is 400. So, (25/2)*400 is (25*400)/2 = 10,000/2 = 5,000. Then subtracting the lower limit, which is 0, so this term is 5,000.Third term: ( int -0.5t² , dt )The integral of -0.5t² is (-0.5/3)t³, which simplifies to (-1/6)t³. Evaluating this from 0 to 20:(-1/6)*(20)³ - (-1/6)*(0)³. Let's compute (20)³ first: 20*20=400, 400*20=8,000. So, (-1/6)*8,000 = -8,000/6 ≈ -1,333.333. The lower limit is 0, so this term is approximately -1,333.333.Now, adding up all three terms:First term: 1,000Second term: 5,000Third term: -1,333.333Total integral = 1,000 + 5,000 - 1,333.333 = Let's compute that.1,000 + 5,000 is 6,000. Then, 6,000 - 1,333.333 is 4,666.667.So, approximately 4,666.667 photographs in total. Since we can't have a fraction of a photo, maybe we should round this. But since the integral gives an exact value, perhaps we can express it as a fraction.Wait, let me check my calculations again because I approximated the third term. Let me do it more precisely.Third term: (-1/6)*8,000 = -8,000/6. Let's divide 8,000 by 6:6 goes into 8 once, remainder 2. 6 goes into 20 three times, remainder 2. 6 goes into 20 three times again, remainder 2. So, 8,000 divided by 6 is 1,333 and 2/6, which simplifies to 1,333 and 1/3, or approximately 1,333.333. So, the exact value is -1,333 and 1/3.So, adding up:1,000 + 5,000 = 6,0006,000 - 1,333 1/3 = Let's convert 6,000 to thirds: 6,000 = 18,000/31,333 1/3 = 4,000/3So, 18,000/3 - 4,000/3 = 14,000/3 ≈ 4,666.666...So, exactly, it's 14,000/3, which is approximately 4,666.666... So, 4,666 and 2/3 photographs.But since the number of photographs should be a whole number, maybe the integral is giving an average or something. Hmm, but actually, the integral represents the total number of photos over the 20 years, so it's okay if it's a fractional number because it's the accumulation over time, not the number per year. So, maybe we can just present it as 14,000/3 or approximately 4,666.67.Wait, but let me think again. The function f(t) is the number of photos taken each year, so integrating it over 20 years gives the total number of photos. So, yes, it's correct that it can be a fractional number because it's the sum over a continuous function. So, 14,000/3 is about 4,666.666..., so I can write it as 4,666.67 or keep it as a fraction.But let me check my integration steps again to make sure I didn't make a mistake.First term: 50t from 0 to 20: 50*20 - 50*0 = 1,000. Correct.Second term: (25/2)t² from 0 to 20: (25/2)*(400) - 0 = 25*200 = 5,000. Correct.Third term: (-1/6)t³ from 0 to 20: (-1/6)*(8,000) - 0 = -1,333.333... Correct.So, adding them up: 1,000 + 5,000 = 6,000; 6,000 - 1,333.333 = 4,666.666... So, that's correct.So, the total number of photographs is 14,000/3, which is approximately 4,666.67.Wait, 14,000 divided by 3 is 4,666.666..., yes. So, I think that's the answer for the first part.Now, moving on to the second part: The photographer wants to display the photos in a gallery, dedicating a wall space proportional to the number of photos taken each year. The wall space for each year t is given by A(t) = k * f(t) square feet, and the total wall space is 1,500 square feet. I need to find the value of k.So, essentially, each year's photos are assigned a wall space of k times the number of photos that year. So, the total wall space is the sum of A(t) from t=0 to t=20, which should equal 1,500.But wait, actually, since f(t) is a continuous function, and we're dealing with wall space proportional to the number of photos each year, which is also a continuous function, the total wall space would be the integral of A(t) from t=0 to t=20, right? Because A(t) is k*f(t), so the total wall space is the integral of k*f(t) dt from 0 to 20.But wait, hold on. Is A(t) the wall space for each year t, meaning that for each year, the wall space is k*f(t). So, if we think of it as a continuous function, the total wall space would be the integral of A(t) over the 20 years, which is k times the integral of f(t) from 0 to 20.But wait, in the first part, we computed the integral of f(t) from 0 to 20 as 14,000/3. So, the total wall space is k*(14,000/3) = 1,500. So, we can solve for k.Let me write that down:Total wall space = ∫₀²⁰ A(t) dt = ∫₀²⁰ k*f(t) dt = k*∫₀²⁰ f(t) dt = k*(14,000/3) = 1,500So, solving for k:k = 1,500 / (14,000/3) = 1,500 * (3/14,000) = (1,500*3)/14,000 = 4,500/14,000Simplify that fraction:Divide numerator and denominator by 50: 4,500 ÷ 50 = 90; 14,000 ÷ 50 = 280. So, 90/280.Simplify further by dividing numerator and denominator by 10: 9/28.So, k = 9/28.Let me check that:9/28 is approximately 0.3214.So, 9/28 * (14,000/3) = (9/28)*(14,000/3) = (9*14,000)/(28*3) = (126,000)/(84) = 1,500. Yes, that works.So, k is 9/28.Wait, but let me think again. Is the wall space for each year t given as A(t) = k*f(t), and the total wall space is the sum over all years? But since f(t) is a continuous function, the total wall space would be the integral of A(t) from 0 to 20, which is k times the integral of f(t) from 0 to 20, which we found to be 14,000/3. So, setting that equal to 1,500 gives k = 9/28.Alternatively, if we were to think of it as a sum rather than an integral, but since t is a continuous variable here, it's more appropriate to use the integral. So, I think my approach is correct.So, to recap:1. Total photos = ∫₀²⁰ f(t) dt = 14,000/3 ≈ 4,666.672. Total wall space = k * (14,000/3) = 1,500 ⇒ k = 9/28So, I think that's the solution.Final Answer1. The total number of photographs taken over 20 years is boxed{dfrac{14000}{3}}.2. The value of ( k ) is boxed{dfrac{9}{28}}.</think>

🧠:<think>Okay, so I have this problem about bacterial growth following a logistic model. My younger relative is working on this project, and they need to figure out the population over time and how modifying the growth rate affects the time to reach half the carrying capacity. Let me try to work through this step by step.First, the logistic growth equation is given as ( frac{dP}{dt} = rP left(1 - frac{P}{K}right) ). I remember that this is a differential equation, and I need to solve it to find ( P(t) ). The initial condition is ( P(0) = P_0 ).I think the standard solution to the logistic equation is something like ( P(t) = frac{K}{1 + left(frac{K - P_0}{P_0}right)e^{-rt}} ). But let me try to derive it properly to make sure.So, starting with the differential equation:( frac{dP}{dt} = rP left(1 - frac{P}{K}right) )This is a separable equation, so I can rewrite it as:( frac{dP}{P left(1 - frac{P}{K}right)} = r dt )To integrate both sides, I need to handle the left side. Let me use partial fractions for the integrand. Let me set:( frac{1}{P left(1 - frac{P}{K}right)} = frac{A}{P} + frac{B}{1 - frac{P}{K}} )Multiplying both sides by ( P left(1 - frac{P}{K}right) ):( 1 = A left(1 - frac{P}{K}right) + B P )Expanding:( 1 = A - frac{A}{K} P + B P )Grouping terms:( 1 = A + left( B - frac{A}{K} right) P )Since this must hold for all P, the coefficients must be equal on both sides. So,For the constant term: ( A = 1 )For the P term: ( B - frac{A}{K} = 0 ) => ( B = frac{A}{K} = frac{1}{K} )So, the partial fractions decomposition is:( frac{1}{P left(1 - frac{P}{K}right)} = frac{1}{P} + frac{1}{K left(1 - frac{P}{K}right)} )Therefore, the integral becomes:( int left( frac{1}{P} + frac{1}{K left(1 - frac{P}{K}right)} right) dP = int r dt )Let me compute each integral separately.First integral: ( int frac{1}{P} dP = ln |P| + C )Second integral: Let me substitute ( u = 1 - frac{P}{K} ), so ( du = -frac{1}{K} dP ) => ( dP = -K du )So, ( int frac{1}{K u} (-K du) = - int frac{1}{u} du = -ln |u| + C = -ln left| 1 - frac{P}{K} right| + C )Putting it all together:( ln |P| - ln left| 1 - frac{P}{K} right| = rt + C )Simplify the left side:( ln left( frac{P}{1 - frac{P}{K}} right) = rt + C )Exponentiate both sides:( frac{P}{1 - frac{P}{K}} = e^{rt + C} = e^{C} e^{rt} )Let me denote ( e^{C} ) as another constant, say ( C' ). So,( frac{P}{1 - frac{P}{K}} = C' e^{rt} )Now, solve for P:Multiply both sides by ( 1 - frac{P}{K} ):( P = C' e^{rt} left( 1 - frac{P}{K} right) )Expand the right side:( P = C' e^{rt} - frac{C'}{K} e^{rt} P )Bring the term with P to the left:( P + frac{C'}{K} e^{rt} P = C' e^{rt} )Factor out P:( P left( 1 + frac{C'}{K} e^{rt} right) = C' e^{rt} )Solve for P:( P = frac{C' e^{rt}}{1 + frac{C'}{K} e^{rt}} )Simplify the denominator:( P = frac{C' e^{rt}}{1 + frac{C'}{K} e^{rt}} = frac{K C' e^{rt}}{K + C' e^{rt}} )Now, apply the initial condition ( P(0) = P_0 ). At t=0:( P_0 = frac{K C'}{K + C'} )Solve for C':Multiply both sides by ( K + C' ):( P_0 (K + C') = K C' )Expand:( P_0 K + P_0 C' = K C' )Bring terms with C' to one side:( P_0 K = K C' - P_0 C' = C' (K - P_0) )Thus,( C' = frac{P_0 K}{K - P_0} )Substitute back into the expression for P(t):( P(t) = frac{K cdot frac{P_0 K}{K - P_0} e^{rt}}{K + frac{P_0 K}{K - P_0} e^{rt}} )Simplify numerator and denominator:Numerator: ( frac{K^2 P_0}{K - P_0} e^{rt} )Denominator: ( K + frac{K P_0}{K - P_0} e^{rt} = K left( 1 + frac{P_0}{K - P_0} e^{rt} right) )So,( P(t) = frac{ frac{K^2 P_0}{K - P_0} e^{rt} }{ K left( 1 + frac{P_0}{K - P_0} e^{rt} right) } = frac{ K P_0 e^{rt} }{ (K - P_0) + P_0 e^{rt} } )Factor numerator and denominator:( P(t) = frac{ K P_0 e^{rt} }{ K - P_0 + P_0 e^{rt} } )Alternatively, factor out ( P_0 ) in the denominator:( P(t) = frac{ K P_0 e^{rt} }{ K - P_0 + P_0 e^{rt} } = frac{ K }{ frac{K - P_0}{P_0} e^{-rt} + 1 } )Which can be written as:( P(t) = frac{ K }{ 1 + left( frac{K - P_0}{P_0} right) e^{-rt} } )Yes, that looks familiar. So that's the expression for ( P(t) ). So that answers the first part.Now, moving on to the second part. The relative genetically modifies the bacteria to enhance their growth rate, changing ( r ) to ( r' ). The carrying capacity ( K ) remains the same. We need to calculate the time ( t ) it takes for the population to reach half of the carrying capacity ( frac{K}{2} ) with the new growth rate ( r' ). It's given that ( P_0 ) is significantly smaller than ( K ), so ( P_0 ll K ). That probably means we can approximate ( P_0 ) as negligible compared to K, but let's see.So, we need to find ( t ) such that ( P(t) = frac{K}{2} ). Using the logistic equation solution:( frac{K}{2} = frac{ K }{ 1 + left( frac{K - P_0}{P_0} right) e^{-r' t} } )Simplify this equation:Multiply both sides by the denominator:( frac{K}{2} left( 1 + left( frac{K - P_0}{P_0} right) e^{-r' t} right) = K )Divide both sides by K:( frac{1}{2} left( 1 + left( frac{K - P_0}{P_0} right) e^{-r' t} right) = 1 )Multiply both sides by 2:( 1 + left( frac{K - P_0}{P_0} right) e^{-r' t} = 2 )Subtract 1:( left( frac{K - P_0}{P_0} right) e^{-r' t} = 1 )Solve for ( e^{-r' t} ):( e^{-r' t} = frac{P_0}{K - P_0} )Take natural logarithm on both sides:( -r' t = ln left( frac{P_0}{K - P_0} right) )Multiply both sides by -1:( r' t = - ln left( frac{P_0}{K - P_0} right) = ln left( frac{K - P_0}{P_0} right) )Thus,( t = frac{1}{r'} ln left( frac{K - P_0}{P_0} right) )But since ( P_0 ) is significantly smaller than ( K ), ( K - P_0 approx K ). So,( frac{K - P_0}{P_0} approx frac{K}{P_0} )Therefore, the time simplifies to:( t approx frac{1}{r'} ln left( frac{K}{P_0} right) )So, that's the time it takes to reach half the carrying capacity with the new growth rate ( r' ).Wait, let me double-check the steps. Starting from ( P(t) = K/2 ):( frac{K}{2} = frac{ K }{ 1 + left( frac{K - P_0}{P_0} right) e^{-r' t} } )Multiply both sides by denominator:( frac{K}{2} left( 1 + left( frac{K - P_0}{P_0} right) e^{-r' t} right) = K )Divide by K:( frac{1}{2} left( 1 + left( frac{K - P_0}{P_0} right) e^{-r' t} right) = 1 )Multiply by 2:( 1 + left( frac{K - P_0}{P_0} right) e^{-r' t} = 2 )Subtract 1:( left( frac{K - P_0}{P_0} right) e^{-r' t} = 1 )So,( e^{-r' t} = frac{P_0}{K - P_0} )Take ln:( -r' t = ln left( frac{P_0}{K - P_0} right) )Multiply by -1:( r' t = ln left( frac{K - P_0}{P_0} right) )Thus,( t = frac{1}{r'} ln left( frac{K - P_0}{P_0} right) )Since ( P_0 ll K ), ( K - P_0 approx K ), so:( t approx frac{1}{r'} ln left( frac{K}{P_0} right) )Yes, that seems correct. So, the time to reach half the carrying capacity is inversely proportional to the growth rate ( r' ) and depends on the natural log of the ratio ( K/P_0 ).So, summarizing:1. The expression for ( P(t) ) is ( frac{K}{1 + left( frac{K - P_0}{P_0} right) e^{-rt}} ).2. The time to reach ( K/2 ) with the new growth rate ( r' ) is approximately ( frac{1}{r'} ln left( frac{K}{P_0} right) ).I think that's it. Let me just check if the approximation is valid. Since ( P_0 ll K ), ( K - P_0 ) is almost K, so the approximation holds. Therefore, the time is roughly ( frac{1}{r'} ln(K/P_0) ).Final Answer1. The population at time ( t ) is ( boxed{P(t) = dfrac{K}{1 + left( dfrac{K - P_0}{P_0} right) e^{-rt}}} ).2. The time to reach half the carrying capacity is ( boxed{t = dfrac{1}{r'} lnleft( dfrac{K}{P_0} right)} ).</think>

🧠:<think>Alright, so I have this problem about a coffee shop owner who's noticed a decline in foot traffic because of graffiti. The owner wants to figure out if the decrease is statistically significant and also wants to know how many additional customers they need to break even on a graffiti removal initiative. Let me try to work through this step by step.First, problem 1: They want to calculate the probability that the mean foot traffic over the last 30 days would be 95 or fewer if the true mean remained at 120. Then, assess if this decrease is statistically significant at the 5% level.Okay, so I remember that when dealing with sample means, we can use the Central Limit Theorem. The foot traffic is normally distributed with a mean of 120 and a standard deviation of 15. The sample size is 30 days, so n=30.I think I need to calculate the z-score for the sample mean of 95. The formula for the z-score when dealing with sample means is:z = (x̄ - μ) / (σ / sqrt(n))Where:- x̄ is the sample mean (95)- μ is the population mean (120)- σ is the population standard deviation (15)- n is the sample size (30)Plugging in the numbers:z = (95 - 120) / (15 / sqrt(30))Let me compute that. First, 95 - 120 is -25. Then, sqrt(30) is approximately 5.477. So, 15 divided by 5.477 is roughly 2.738. Then, -25 divided by 2.738 is approximately -9.13.Wait, that seems like a really low z-score. Is that right? Let me double-check my calculations.95 - 120 is indeed -25. sqrt(30) is about 5.477. 15 divided by 5.477 is approximately 2.738. So, -25 / 2.738 is roughly -9.13. Hmm, that seems correct. So, the z-score is about -9.13.Now, to find the probability that the mean foot traffic is 95 or fewer, we need the area to the left of z = -9.13 in the standard normal distribution. But wait, a z-score of -9.13 is way beyond the typical tables. I know that z-scores beyond about -3 or -4 are already extremely rare, so a z-score of -9.13 would have a probability practically zero.But let me see if I can compute it or at least approximate it. Using a standard normal distribution table, the probability for z = -3 is about 0.13%, and it decreases exponentially as z becomes more negative. So, for z = -9.13, the probability is effectively zero. So, the p-value here is less than 0.0001.Since the p-value is way below the 5% significance level (0.05), we can reject the null hypothesis that the true mean foot traffic remains at 120. Therefore, the decrease is statistically significant.Wait, but just to make sure, is the alternative hypothesis one-tailed or two-tailed? Since the owner believes the graffiti has caused a decrease, we're only concerned with the lower tail. So, it's a one-tailed test. Therefore, the p-value is the area to the left of z = -9.13, which is practically zero, so definitely less than 0.05. So, yes, statistically significant.Moving on to problem 2: The owner wants to invest 500 in a community initiative to remove graffiti, hoping to increase foot traffic back to at least 110 people per day. Each additional customer spends an average of 5. We need to calculate the expected number of additional customers needed per day over the next 30 days to break even on this investment.So, breaking even means that the revenue generated from the additional customers should cover the cost of the initiative. The cost is 500, and each additional customer contributes 5. So, the number of additional customers needed to break even can be calculated as:Number of customers = Total cost / Revenue per customerWhich is 500 / 5 = 100 customers.But wait, that's over the next 30 days. So, per day, the number of additional customers needed is 100 / 30 ≈ 3.333 customers per day.But the problem says the increase in foot traffic is uniformly distributed over the 30 days. So, does that mean the total increase over 30 days is 100 customers? Or is it 100 customers per day?Wait, let me read the problem again: "calculate the expected number of additional customers needed per day over the next 30 days to break even on this investment."So, the total cost is 500. Each additional customer brings in 5. So, total additional revenue needed is 500. Therefore, total additional customers needed over 30 days is 500 / 5 = 100 customers. Therefore, per day, it's 100 / 30 ≈ 3.333 customers per day.But since we can't have a fraction of a customer, we might need to round up. But the problem says to calculate the expected number, so we can keep it as a decimal.Wait, but hold on. The owner wants to increase foot traffic back to at least 110 people per day. Currently, the foot traffic is 95 per day. So, the increase needed per day is 110 - 95 = 15 customers per day.But wait, is that the case? Or is the 110 the target, regardless of the current level?Wait, the problem says: "hoping to increase foot traffic back to at least 110 people per day." So, the target is 110 per day. The current average is 95, so the increase needed is 15 per day.But the break-even calculation is separate. The owner is investing 500, and each additional customer brings in 5. So, regardless of the target, the break-even is 100 customers over 30 days, which is about 3.333 per day.But wait, maybe I'm mixing two things here. The target is 110 per day, but the break-even is based on the investment. So, perhaps the owner wants to know how many additional customers per day are needed beyond the current level to cover the 500 cost.So, current foot traffic is 95 per day. If they increase to 110, that's an increase of 15 per day. The revenue from these additional customers would be 15 * 5 = 75 per day. Over 30 days, that's 75 * 30 = 2250. But the investment is only 500, so actually, they would make a profit. But the question is about breaking even, not necessarily reaching the target.Wait, the problem says: "calculate the expected number of additional customers needed per day over the next 30 days to break even on this investment."So, regardless of the target, they need to find how many additional customers per day are needed so that the total revenue from these customers equals 500.So, total revenue needed: 500.Each customer contributes 5, so total customers needed: 500 / 5 = 100 customers over 30 days.Therefore, per day: 100 / 30 ≈ 3.333 customers per day.So, approximately 3.33 additional customers per day.But the problem says "the increase in foot traffic is uniformly distributed over the 30 days." So, does that affect the calculation? I think it just means that the increase is spread out evenly, so we don't have to worry about variability in the distribution; it's a uniform increase. So, our calculation remains the same.Therefore, the expected number of additional customers needed per day is approximately 3.33.But let me make sure. Is there another way to interpret this? Maybe the owner wants to reach at least 110 per day, so the increase is 15 per day, which would bring in 15 * 5 = 75 per day. Over 30 days, that's 2250, which is more than the 500 investment. So, in that case, they would not only break even but make a profit.But the question is specifically about breaking even, not necessarily reaching the target. So, perhaps the target is just a hope, but the break-even is a separate calculation.So, to break even, they need 100 additional customers over 30 days, which is about 3.33 per day.Therefore, the answer is approximately 3.33 additional customers per day.Wait, but let me think again. If the foot traffic is currently 95 per day, and they want to increase it to 110, that's an increase of 15 per day. So, over 30 days, that's 450 additional customers. Each brings in 5, so total revenue is 450 * 5 = 2250. So, the investment is 500, so the profit is 2250 - 500 = 1750.But the question is about breaking even, not about reaching the target. So, breaking even is when the revenue from additional customers equals the investment. So, regardless of the target, how many additional customers are needed to cover 500.So, that's 100 customers over 30 days, which is about 3.33 per day.Therefore, the expected number is 100 / 30 ≈ 3.33.So, summarizing:1. The probability is practically zero, so the decrease is statistically significant.2. They need approximately 3.33 additional customers per day to break even.But let me write the exact calculations.For problem 1:z = (95 - 120) / (15 / sqrt(30)) = (-25) / (15 / 5.477) ≈ -25 / 2.738 ≈ -9.13The p-value is P(Z ≤ -9.13), which is effectively 0. So, p < 0.0001, which is less than 0.05. Therefore, statistically significant.For problem 2:Total cost: 500Revenue per customer: 5Total customers needed: 500 / 5 = 100Per day: 100 / 30 ≈ 3.333So, approximately 3.33 additional customers per day.I think that's it.</think>