🧠:<think>Okay, so I have these two problems to solve related to Dr. Emily Hart's findings. Let me tackle them one by one.Starting with the first problem: It's about modeling the production of widgets over time using a differential equation. The equation given is a logistic growth model, right? The differential equation is:[ frac{dP}{dt} = kPleft(1 - frac{P}{M}right) ]Where ( P(t) ) is the production, ( k ) is the growth rate, ( M ) is the maximum capacity, and ( t ) is time in months. The initial condition is ( P(0) = 100 ) widgets, and the maximum capacity ( M ) is 1000 widgets. The growth rate ( k ) is 0.1.I remember that the solution to the logistic equation is:[ P(t) = frac{M}{1 + left(frac{M - P_0}{P_0}right)e^{-kM t}} ]Wait, let me verify that. So, the general solution for the logistic equation is:[ P(t) = frac{M}{1 + left(frac{M - P_0}{P_0}right)e^{-kt}} ]Yes, that's correct. So, plugging in the given values:( P_0 = 100 ), ( M = 1000 ), ( k = 0.1 ).So, substituting these into the formula:First, compute ( frac{M - P_0}{P_0} ):( frac{1000 - 100}{100} = frac{900}{100} = 9 ).So, the equation becomes:[ P(t) = frac{1000}{1 + 9e^{-0.1 t}} ]Let me double-check that. If I plug ( t = 0 ), I should get ( P(0) = 100 ):[ P(0) = frac{1000}{1 + 9e^{0}} = frac{1000}{1 + 9} = frac{1000}{10} = 100 ]Yes, that works. So, that seems correct.Alright, moving on to the second problem. It involves calculating the total wages paid over the first 5 years. The wage function is given as:[ W(t) = W_0 e^{rt} ]Where ( W_0 = 50 ) dollars, and ( r = 0.02 ) per month. So, we need to find the total wages over 5 years. Since the time is in months, 5 years would be 60 months.Total wages would be the integral of ( W(t) ) from ( t = 0 ) to ( t = 60 ). So, the integral of ( W(t) ) with respect to ( t ) is:[ int_{0}^{60} W_0 e^{rt} dt ]Let me compute that. The integral of ( e^{rt} ) is ( frac{1}{r} e^{rt} ). So,[ int_{0}^{60} 50 e^{0.02 t} dt = 50 times left[ frac{1}{0.02} e^{0.02 t} right]_0^{60} ]Simplify:First, ( frac{1}{0.02} = 50 ). So,[ 50 times 50 times left( e^{0.02 times 60} - e^{0} right) ]Compute ( 0.02 times 60 = 1.2 ). So,[ 2500 times (e^{1.2} - 1) ]Now, compute ( e^{1.2} ). I know that ( e^1 ) is approximately 2.71828, and ( e^{0.2} ) is approximately 1.2214. So, multiplying these:( e^{1.2} = e^{1 + 0.2} = e^1 times e^{0.2} approx 2.71828 times 1.2214 approx 3.3201 ).So, ( e^{1.2} - 1 approx 3.3201 - 1 = 2.3201 ).Therefore, total wages:( 2500 times 2.3201 approx 2500 times 2.3201 ).Compute 2500 * 2 = 5000, 2500 * 0.3201 = 2500 * 0.3 = 750, 2500 * 0.0201 = 50.25. So, total:750 + 50.25 = 800.25.So, total is 5000 + 800.25 = 5800.25.Wait, that seems a bit high. Let me check my calculations again.Wait, 2500 * 2.3201:Break it down:2.3201 * 2500.First, 2 * 2500 = 5000.0.3201 * 2500:0.3 * 2500 = 750.0.0201 * 2500 = 50.25.So, 750 + 50.25 = 800.25.So, total is 5000 + 800.25 = 5800.25.So, approximately 5,800.25.But wait, that's over 5 years, right? So, that seems plausible.But let me check the integral again.The integral of ( W(t) ) is ( frac{W_0}{r} (e^{rt} - 1) ). So, substituting:( frac{50}{0.02} (e^{0.02 * 60} - 1) = 2500 (e^{1.2} - 1) approx 2500 * 2.3201 approx 5800.25 ).Yes, that seems correct.Alternatively, if I compute ( e^{1.2} ) more accurately, maybe I can get a better approximation.Using a calculator, ( e^{1.2} ) is approximately 3.32011692277.So, 3.32011692277 - 1 = 2.32011692277.Multiply by 2500:2.32011692277 * 2500.2 * 2500 = 5000.0.32011692277 * 2500 = 800.292306925.So, total is 5000 + 800.292306925 ≈ 5800.2923.So, approximately 5,800.29.So, rounding to the nearest cent, it's 5,800.29.But since the question says "calculate the total wages", it might expect an exact expression or a numerical value. Since they gave us numbers with two decimal places, probably expects a numerical answer.So, I think 5,800.29 is the total wages over 5 years.Wait, but 5 years is 60 months, right? So, the time period is correct.Alternatively, if we were to compute it as a sum of monthly wages, but since the wage function is continuous, integrating is the correct approach.So, I think that's the right answer.Final Answer1. The production function is boxed{P(t) = dfrac{1000}{1 + 9e^{-0.1t}}}.2. The total wages paid over the first 5 years is boxed{5800.29} dollars.</think>

🧠:<think>Okay, so I have this problem about dividing an inheritance, and I need to figure out how much each beneficiary gets. Let me try to break it down step by step.First, the estate is worth 5 million. It's being divided among three parties: two children and a charitable trust. The charitable trust gets a fixed percentage, which is 15%. Then, the remaining amount is split between the two children based on their ages. The older child is twice as old as the younger one. Hmm, okay.Let me write down what I know:- Total estate: 5,000,000- Charitable trust gets 15%- Remaining amount is split between two children proportionally to their ages- Older child's age is twice the younger child's ageSo, first, I need to calculate how much the charitable trust receives. That should be straightforward: 15% of 5 million.Let me compute that:15% of 5,000,000 = 0.15 * 5,000,000 = 750,000.So, the charitable trust gets 750,000.Now, the remaining amount is 5,000,000 - 750,000 = 4,250,000.This 4,250,000 needs to be divided between the two children based on their ages. The older child is twice as old as the younger one. So, if I let the younger child's age be 'x' years, then the older child's age is '2x' years.Therefore, the ratio of their ages is x : 2x, which simplifies to 1:2.That means the younger child gets 1 part, and the older child gets 2 parts of the remaining 4,250,000.So, the total number of parts is 1 + 2 = 3 parts.Each part is equal to 4,250,000 divided by 3.Let me calculate that:4,250,000 / 3 ≈ 1,416,666.67 per part.Therefore, the younger child gets 1 part, which is approximately 1,416,666.67, and the older child gets 2 parts, which is approximately 2,833,333.33.Wait, let me check if that adds up correctly. 1,416,666.67 + 2,833,333.33 = 4,250,000, which matches the remaining amount after the charitable trust. So that seems correct.So, summarizing:- Charitable trust: 750,000- Younger child: ~1,416,666.67- Older child: ~2,833,333.33But the problem says to calculate the exact amount, so maybe I should represent it as fractions instead of decimals to keep it precise.So, 4,250,000 divided by 3 is equal to (4,250,000 / 3) = 1,416,666 and 2/3 dollars. So, the exact amount for the younger child is 1,416,666.666..., and the older child is twice that, which is 2,833,333.333...Alternatively, in fractions, that's 1,416,666 2/3 and 2,833,333 1/3.So, that's part 1 done.Now, moving on to part 2. The attorney also has to handle guardianship matters. The older child is currently 16 and will manage their portion upon turning 18. The growth rate is modeled by the exponential function A(t) = A0 * e^(rt), where A0 is the initial amount, r is the annual growth rate, and t is time in years. The growth rate is 5% per annum.So, I need to calculate the value of the older child's portion at age 18. Since the older child is currently 16, that means the time until they can manage their portion is 2 years.So, t = 2 years.Given that, let's note down the values:- A0: the initial amount, which is the older child's portion from part 1, which is approximately 2,833,333.33- r: 5% per annum, which is 0.05- t: 2 yearsSo, plugging into the formula:A(t) = A0 * e^(r*t)So, A(2) = 2,833,333.33 * e^(0.05*2)First, let me compute the exponent: 0.05 * 2 = 0.10So, e^0.10. I need to calculate e raised to the power of 0.10.I remember that e^0.1 is approximately 1.105170918.So, e^0.10 ≈ 1.105170918Therefore, A(2) ≈ 2,833,333.33 * 1.105170918Let me compute that.First, let me write 2,833,333.33 as a fraction to be precise. Since 2,833,333.33 is 2,833,333 and 1/3 dollars, which is equal to 8,500,000/3.Wait, 2,833,333.33 * 3 = 8,500,000, so yes, 2,833,333.33 = 8,500,000 / 3.So, A(2) = (8,500,000 / 3) * 1.105170918Let me compute 8,500,000 * 1.105170918 first.8,500,000 * 1.105170918 = ?Let me compute 8,500,000 * 1 = 8,500,0008,500,000 * 0.105170918 ≈ 8,500,000 * 0.105170918Compute 8,500,000 * 0.1 = 850,0008,500,000 * 0.005170918 ≈ 8,500,000 * 0.005 = 42,500So, approximately, 850,000 + 42,500 = 892,500Therefore, total is approximately 8,500,000 + 892,500 = 9,392,500But wait, that's an approximation. Let me compute it more accurately.Compute 8,500,000 * 0.105170918:First, 8,500,000 * 0.1 = 850,0008,500,000 * 0.005170918:Compute 8,500,000 * 0.005 = 42,5008,500,000 * 0.000170918 ≈ 8,500,000 * 0.00017 = 1,445So, adding up: 42,500 + 1,445 = 43,945Therefore, total 850,000 + 43,945 = 893,945So, 8,500,000 * 1.105170918 ≈ 8,500,000 + 893,945 = 9,393,945Now, divide that by 3:9,393,945 / 3 ≈ 3,131,315Wait, let me compute that division precisely.9,393,945 divided by 3:3 goes into 9 three times, 3*3=9, remainder 0.Bring down 3: 03. 3 goes into 3 once, 1*3=3, remainder 0.Bring down 9: 09. 3 goes into 9 three times, 3*3=9, remainder 0.Bring down 3: 03. 3 goes into 3 once, 1*3=3, remainder 0.Bring down 9: 09. 3 goes into 9 three times, 3*3=9, remainder 0.Bring down 4: 04. 3 goes into 4 once, 1*3=3, remainder 1.Bring down 5: 15. 3 goes into 15 five times, 5*3=15, remainder 0.So, putting it all together: 3,131,315.Wait, let me verify:3 * 3,131,315 = 9,393,945. Yes, that's correct.So, A(2) ≈ 3,131,315.But let me check if I can compute it more accurately without approximating e^0.10.Alternatively, I can use the exact value of e^0.10, which is approximately 1.1051709180756477.So, 2,833,333.33 * 1.1051709180756477Let me compute 2,833,333.33 * 1.1051709180756477First, let me write 2,833,333.33 as 2,833,333 + 0.333333...So, 2,833,333 * 1.1051709180756477 + 0.333333... * 1.1051709180756477Compute the first part:2,833,333 * 1.1051709180756477Let me compute 2,833,333 * 1 = 2,833,3332,833,333 * 0.1051709180756477 ≈ ?Compute 2,833,333 * 0.1 = 283,333.32,833,333 * 0.0051709180756477 ≈ ?Compute 2,833,333 * 0.005 = 14,166.6652,833,333 * 0.0001709180756477 ≈ ?Compute 2,833,333 * 0.0001 = 283.33332,833,333 * 0.0000709180756477 ≈ 2,833,333 * 0.00007 ≈ 198.33331So, adding up:283,333.3 + 14,166.665 + 283.33331 + 198.33331 ≈283,333.3 + 14,166.665 = 297,500297,500 + 283.33331 ≈ 297,783.33331297,783.33331 + 198.33331 ≈ 297,981.66662So, total ≈ 2,833,333 + 297,981.66662 ≈ 3,131,314.66662Now, compute the second part: 0.333333... * 1.1051709180756477 ≈ 0.368390306So, total A(2) ≈ 3,131,314.66662 + 0.368390306 ≈ 3,131,315.035So, approximately 3,131,315.04Therefore, the value of the older child's portion at age 18 is approximately 3,131,315.04.But let me see if I can represent this more precisely. Since we have A0 as an exact fraction, 8,500,000/3, and e^0.10 is an irrational number, so we can't represent it exactly. So, the answer will be approximate.Alternatively, if I use more decimal places for e^0.10, I can get a more accurate result.e^0.10 ≈ 1.1051709180756477So, 2,833,333.33 * 1.1051709180756477Let me compute this using a calculator approach.First, 2,833,333.33 * 1 = 2,833,333.332,833,333.33 * 0.1051709180756477 ≈ ?Compute 2,833,333.33 * 0.1 = 283,333.3332,833,333.33 * 0.0051709180756477 ≈ ?Compute 2,833,333.33 * 0.005 = 14,166.666652,833,333.33 * 0.0001709180756477 ≈ ?Compute 2,833,333.33 * 0.0001 = 283.33333332,833,333.33 * 0.0000709180756477 ≈ 2,833,333.33 * 0.00007 ≈ 198.3333333So, adding up:283,333.333 + 14,166.66665 + 283.3333333 + 198.3333333 ≈283,333.333 + 14,166.66665 = 297,500297,500 + 283.3333333 ≈ 297,783.3333333297,783.3333333 + 198.3333333 ≈ 297,981.6666666So, total ≈ 2,833,333.33 + 297,981.6666666 ≈ 3,131,315.0Adding the decimal parts, it's approximately 3,131,315.0So, rounding to the nearest cent, it's 3,131,315.00Wait, but earlier I had 3,131,315.04, but with more precise calculation, it's about 3,131,315.00.Hmm, maybe the difference comes from the approximation in e^0.10.Alternatively, perhaps I should use the formula with the exact initial amount as a fraction.So, A0 = 8,500,000 / 3So, A(t) = (8,500,000 / 3) * e^(0.10)Compute e^0.10 ≈ 1.1051709180756477So, A(t) = (8,500,000 / 3) * 1.1051709180756477Compute 8,500,000 * 1.1051709180756477 ≈ 9,393,945.757143403Then, divide by 3: 9,393,945.757143403 / 3 ≈ 3,131,315.252381134So, approximately 3,131,315.25So, rounding to the nearest cent, it's 3,131,315.25Therefore, the value of the older child's portion at age 18 is approximately 3,131,315.25So, to summarize:1. The charitable trust receives 750,000, the younger child receives approximately 1,416,666.67, and the older child receives approximately 2,833,333.33.2. The older child's portion grows to approximately 3,131,315.25 by the time they turn 18.I think that's all. Let me just double-check my calculations to make sure I didn't make any errors.For part 1:- 15% of 5 million is indeed 750,000.- Remaining is 4,250,000.- Ratio of ages is 1:2, so total parts 3.- 4,250,000 / 3 is approximately 1,416,666.67 and 2,833,333.33.Yes, that seems correct.For part 2:- Older child is 16, so t=2.- A0 is 2,833,333.33.- Growth rate 5%, so r=0.05.- A(t) = 2,833,333.33 * e^(0.10) ≈ 3,131,315.25Yes, that seems correct.I think I've covered all the steps and double-checked the calculations. So, I'm confident in these results.</think>

🧠:<think>Okay, so I have this differential equation to solve: dC/dt = -kC + P(t), where P(t) is A cos(ωt). The initial condition is C(0) = C0. Hmm, I think this is a linear first-order differential equation. I remember that for equations of the form dy/dt + P(t)y = Q(t), we can use an integrating factor. Let me try to rearrange this equation.So, moving the -kC to the left side, I get dC/dt + kC = P(t). Yeah, that looks right. The integrating factor, μ(t), should be e^(∫k dt) which is e^(kt). Multiplying both sides by the integrating factor:e^(kt) dC/dt + k e^(kt) C = e^(kt) P(t)The left side is the derivative of (e^(kt) C) with respect to t. So, integrating both sides from t=0 to t:∫₀ᵗ d/dτ (e^(kτ) C(τ)) dτ = ∫₀ᵗ e^(kτ) P(τ) dτWhich simplifies to:e^(kt) C(t) - e^(0) C(0) = ∫₀ᵗ e^(kτ) A cos(ωτ) dτSo, e^(kt) C(t) = C0 + ∫₀ᵗ e^(kτ) A cos(ωτ) dτTherefore, C(t) = e^(-kt) C0 + e^(-kt) ∫₀ᵗ e^(kτ) A cos(ωτ) dτNow, I need to compute that integral. Let me focus on ∫ e^(kτ) cos(ωτ) dτ. I think integration by parts is the way to go here, or maybe use a standard integral formula.I recall that ∫ e^(at) cos(bt) dt = e^(at)/(a² + b²) (a cos(bt) + b sin(bt)) + CLet me verify that. Let’s differentiate the right side:d/dt [e^(at)/(a² + b²) (a cos(bt) + b sin(bt))] = e^(at)/(a² + b²) [a cos(bt) + b sin(bt)] * a + e^(at)/(a² + b²) [-a b sin(bt) + b² cos(bt)]= e^(at)/(a² + b²) [a² cos(bt) + ab sin(bt) - ab sin(bt) + b² cos(bt)]= e^(at)/(a² + b²) (a² + b²) cos(bt)= e^(at) cos(bt)Yes, that works. So, the integral is e^(at)/(a² + b²) (a cos(bt) + b sin(bt)).In our case, a = k and b = ω. So, the integral from 0 to t is:[e^(kτ)/(k² + ω²) (k cos(ωτ) + ω sin(ωτ))] from 0 to tWhich is [e^(kt)/(k² + ω²) (k cos(ωt) + ω sin(ωt))] - [e^(0)/(k² + ω²) (k cos(0) + ω sin(0))]Simplify that:= e^(kt)/(k² + ω²) (k cos(ωt) + ω sin(ωt)) - (1)/(k² + ω²) (k * 1 + ω * 0)= [e^(kt) (k cos(ωt) + ω sin(ωt)) - k]/(k² + ω²)So, plugging this back into the expression for C(t):C(t) = e^(-kt) C0 + e^(-kt) * [e^(kt) (k cos(ωt) + ω sin(ωt)) - k]/(k² + ω²)Simplify term by term:First term: e^(-kt) C0Second term: e^(-kt) * [e^(kt) (k cos(ωt) + ω sin(ωt)) - k]/(k² + ω²)Let’s split the second term:= [e^(-kt) * e^(kt) (k cos(ωt) + ω sin(ωt))]/(k² + ω²) - [e^(-kt) * k]/(k² + ω²)Simplify e^(-kt) * e^(kt) = 1, so first part is (k cos(ωt) + ω sin(ωt))/(k² + ω²)Second part: -k e^(-kt)/(k² + ω²)So, putting it all together:C(t) = e^(-kt) C0 + (k cos(ωt) + ω sin(ωt))/(k² + ω²) - k e^(-kt)/(k² + ω²)We can factor out e^(-kt) in the first and last terms:= e^(-kt) [C0 - k/(k² + ω²)] + (k cos(ωt) + ω sin(ωt))/(k² + ω²)Alternatively, we can write it as:C(t) = e^(-kt) [C0 - k/(k² + ω²)] + [k cos(ωt) + ω sin(ωt)]/(k² + ω²)I think that's the solution. Let me check if it makes sense. As t approaches infinity, the exponential term should decay, and the crime rate should approach the steady-state solution, which is [k cos(ωt) + ω sin(ωt)]/(k² + ω²). Wait, but that's still oscillatory. Hmm, maybe I made a mistake.Wait, actually, the steady-state solution for a periodic forcing function is typically a particular solution that's also periodic. So, perhaps the solution is correct. The transient term is e^(-kt) times some constant, and the steady-state is the oscillatory part.Okay, moving on to part 2. The administrator wants to minimize the average crime rate over a period T. So, the average crime rate over T is (1/T) ∫₀^T C(t) dt.We need to express this average as an integral and find ω that minimizes it.First, let's write the average:Average C = (1/T) ∫₀^T C(t) dtFrom part 1, C(t) is:C(t) = e^(-kt) [C0 - k/(k² + ω²)] + [k cos(ωt) + ω sin(ωt)]/(k² + ω²)So, integrating term by term:Average C = (1/T) [ ∫₀^T e^(-kt) [C0 - k/(k² + ω²)] dt + ∫₀^T [k cos(ωt) + ω sin(ωt)]/(k² + ω²) dt ]Let me compute each integral separately.First integral: I1 = ∫₀^T e^(-kt) [C0 - k/(k² + ω²)] dtThis is [C0 - k/(k² + ω²)] ∫₀^T e^(-kt) dtThe integral of e^(-kt) dt is (-1/k) e^(-kt). Evaluated from 0 to T:= [C0 - k/(k² + ω²)] [ (-1/k) e^(-kT) + (1/k) e^(0) ]= [C0 - k/(k² + ω²)] [ (1 - e^(-kT))/k ]Second integral: I2 = ∫₀^T [k cos(ωt) + ω sin(ωt)]/(k² + ω²) dtWe can split this into two integrals:= [k/(k² + ω²)] ∫₀^T cos(ωt) dt + [ω/(k² + ω²)] ∫₀^T sin(ωt) dtCompute each:∫ cos(ωt) dt = (1/ω) sin(ωt)∫ sin(ωt) dt = (-1/ω) cos(ωt)So,= [k/(k² + ω²)] [ (1/ω) sin(ωT) - (1/ω) sin(0) ] + [ω/(k² + ω²)] [ (-1/ω) cos(ωT) + (1/ω) cos(0) ]Simplify:= [k/(k² + ω²)] (sin(ωT)/ω) + [ω/(k² + ω²)] ( (-cos(ωT) + 1)/ω )= [k sin(ωT)]/(ω(k² + ω²)) + [ (-cos(ωT) + 1) ]/(k² + ω² )So, putting it all together, the average is:Average C = (1/T)[ I1 + I2 ]= (1/T)[ (C0 - k/(k² + ω²))(1 - e^(-kT))/k + [k sin(ωT)]/(ω(k² + ω²)) + (-cos(ωT) + 1)/(k² + ω²) ]Hmm, this looks a bit complicated. Maybe we can simplify it further.Let me write it step by step:Average C = (1/T) * [ (C0 - k/(k² + ω²))(1 - e^(-kT))/k + (k sin(ωT))/(ω(k² + ω²)) + (1 - cos(ωT))/(k² + ω²) ]Now, to minimize this average with respect to ω, we need to take the derivative of Average C with respect to ω, set it to zero, and solve for ω.But this seems quite involved. Maybe we can make some approximations or consider specific cases.Wait, the problem says to assume A, k, and T are known constants. So, perhaps we can treat ω as a variable and find the ω that minimizes the average.Alternatively, maybe we can consider the average over a period T where T is the period of P(t). Since P(t) is periodic with period 2π/ω, if we take T = 2π/ω, then the integral over T would simplify because the sine and cosine terms would integrate over a full period.Wait, but the problem says "over a period T", so maybe T is fixed, and ω is variable. Hmm, that complicates things because T is fixed, but ω can vary, so the period of P(t) is 2π/ω, which may not align with T.Alternatively, perhaps the administrator is considering the average over one period of the policy intervention, so T = 2π/ω. If that's the case, then T is dependent on ω, but the problem says T is a known constant. Hmm, confusing.Wait, the problem says "the average crime rate over a period T". So, T is fixed, and ω is variable. So, the period of P(t) is 2π/ω, but the average is taken over a fixed interval T, which may not be an integer multiple of the period of P(t). That complicates the integral.Alternatively, maybe the administrator is considering the average over one period of the policy, so T = 2π/ω. Then, T would depend on ω, but the problem states T is a known constant. Hmm, perhaps I need to proceed without assuming T is related to ω.Alternatively, maybe the integral over T can be expressed in terms of ω, and then we can find the ω that minimizes the average.Let me consider that the average is:Average C = (1/T)[ (C0 - k/(k² + ω²))(1 - e^(-kT))/k + (k sin(ωT))/(ω(k² + ω²)) + (1 - cos(ωT))/(k² + ω²) ]To minimize this with respect to ω, we can take the derivative d(Average C)/dω and set it to zero.This will involve differentiating each term with respect to ω. Let me denote the terms as:Term1 = (C0 - k/(k² + ω²))(1 - e^(-kT))/kTerm2 = (k sin(ωT))/(ω(k² + ω²))Term3 = (1 - cos(ωT))/(k² + ω²)So, Average C = (1/T)(Term1 + Term2 + Term3)Compute derivative d(Average C)/dω:= (1/T)( d(Term1)/dω + d(Term2)/dω + d(Term3)/dω )Set this equal to zero.Let me compute each derivative.First, d(Term1)/dω:Term1 = (C0 - k/(k² + ω²))(1 - e^(-kT))/kSo, d(Term1)/dω = [0 - (-2kω)/(k² + ω²)^2 ] * (1 - e^(-kT))/k= [2kω/(k² + ω²)^2 ] * (1 - e^(-kT))/k= 2ω(1 - e^(-kT))/(k² + ω²)^2Second, d(Term2)/dω:Term2 = (k sin(ωT))/(ω(k² + ω²))Let me write this as k sin(ωT) / (ω(k² + ω²)) = k sin(ωT) / (ω(k² + ω²))Let me denote f(ω) = sin(ωT), g(ω) = ω(k² + ω²)So, Term2 = k f(ω)/g(ω)Then, d(Term2)/dω = k [ f’(ω) g(ω) - f(ω) g’(ω) ] / [g(ω)]²Compute f’(ω) = T cos(ωT)g(ω) = ω(k² + ω²)g’(ω) = (k² + ω²) + ω*(2ω) = k² + ω² + 2ω² = k² + 3ω²So,d(Term2)/dω = k [ T cos(ωT) * ω(k² + ω²) - sin(ωT) * (k² + 3ω²) ] / [ω²(k² + ω²)^2 ]Simplify numerator:= k [ T ω(k² + ω²) cos(ωT) - (k² + 3ω²) sin(ωT) ] / [ω²(k² + ω²)^2 ]Third, d(Term3)/dω:Term3 = (1 - cos(ωT))/(k² + ω²)So, d(Term3)/dω = [ T sin(ωT) * (k² + ω²) - (1 - cos(ωT)) * 2ω ] / (k² + ω²)^2Putting it all together, the derivative is:(1/T)[ 2ω(1 - e^(-kT))/(k² + ω²)^2 + k [ T ω(k² + ω²) cos(ωT) - (k² + 3ω²) sin(ωT) ] / [ω²(k² + ω²)^2 ] + [ T sin(ωT) (k² + ω²) - 2ω(1 - cos(ωT)) ] / (k² + ω²)^2 ] = 0This is a very complicated expression. It might be difficult to solve analytically. Perhaps we can consider specific cases or make approximations.Alternatively, maybe we can consider the case where T is very large, so that e^(-kT) is negligible. Then, the term involving (1 - e^(-kT)) becomes approximately (1 - 0) = 1.But the problem doesn't specify that T is large, so maybe that's not the way to go.Alternatively, perhaps we can consider the case where the transient term has decayed, so for t large enough, the crime rate is dominated by the steady-state solution. But again, the problem is about the average over a period T, which may not necessarily be large.Alternatively, maybe we can consider the average over one period of the policy, i.e., T = 2π/ω. Then, the integral over T would simplify because sin and cos terms would integrate over a full period.Let me try that approach. Suppose T = 2π/ω. Then, sin(ωT) = sin(2π) = 0, and cos(ωT) = cos(2π) = 1.So, plugging T = 2π/ω into the average:Average C = (1/T)[ (C0 - k/(k² + ω²))(1 - e^(-kT))/k + 0 + (1 - 1)/(k² + ω²) ]Wait, that simplifies to:= (1/T)[ (C0 - k/(k² + ω²))(1 - e^(-kT))/k + 0 + 0 ]= (1/T)(C0 - k/(k² + ω²))(1 - e^(-kT))/kBut this is only valid if T = 2π/ω. However, the problem states that T is a known constant, so perhaps this approach isn't directly applicable unless we set T = 2π/ω, which would make ω = 2π/T. But then, ω is fixed, which contradicts the idea of minimizing over ω.Alternatively, maybe the administrator is considering the average over one period of the policy, so T = 2π/ω, but since T is given, ω would have to be 2π/T. But then, ω is fixed, so there's nothing to minimize. Hmm, conflicting.Perhaps I need to proceed without assuming T is related to ω. Let me go back to the expression for the average:Average C = (1/T)[ (C0 - k/(k² + ω²))(1 - e^(-kT))/k + (k sin(ωT))/(ω(k² + ω²)) + (1 - cos(ωT))/(k² + ω²) ]To minimize this with respect to ω, we can set the derivative to zero. However, as seen earlier, the derivative is quite complex. Maybe we can consider that the term involving (1 - e^(-kT)) is a constant with respect to ω, so the minimization would mainly depend on the other terms.Alternatively, perhaps we can consider that for large k, the exponential term decays quickly, so the transient term is negligible, and the average is dominated by the steady-state terms. But without knowing the relationship between k and T, it's hard to say.Alternatively, maybe we can consider that the average crime rate is minimized when the amplitude of the steady-state solution is minimized. The steady-state solution is [k cos(ωt) + ω sin(ωt)]/(k² + ω²). The amplitude of this is sqrt(k² + ω²)/(k² + ω²) = 1/sqrt(k² + ω²). So, to minimize the amplitude, we need to maximize sqrt(k² + ω²), which would occur as ω approaches infinity. But that can't be right because higher ω would make the policy oscillate faster, but the average might not necessarily be minimized.Wait, actually, the amplitude of the steady-state solution is 1/sqrt(k² + ω²). So, to minimize the amplitude, we need to maximize sqrt(k² + ω²), which would occur as ω increases. However, the average crime rate also depends on the transient term, which involves e^(-kt). So, perhaps there's a balance.Alternatively, maybe the average crime rate is minimized when the frequency ω is such that the policy interventions are most effective in damping the crime rate. Perhaps when ω is such that the policy oscillates at a frequency that resonates with the natural decay rate k. But I'm not sure.Alternatively, maybe we can consider that the average of the steady-state solution over a period is zero because it's oscillatory. But wait, the average of cos and sin over a period is zero, but in our case, the average is over a fixed T, not necessarily a period of the policy.Wait, if T is a multiple of the period of P(t), then the average of the steady-state solution would be zero. But if T is not a multiple, then the average would have some residual.But in the expression for the average, the terms involving sin(ωT) and (1 - cos(ωT)) suggest that unless T is a multiple of the period, those terms won't cancel out.Hmm, this is getting complicated. Maybe I need to consider that the average crime rate is composed of a transient term and a steady-state term. The transient term decays exponentially, so for large T, the average would be dominated by the steady-state term. But since T is fixed, perhaps the optimal ω is such that the steady-state contribution is minimized.Wait, the steady-state contribution to the average is [k sin(ωT)]/(ω(k² + ω²)) + (1 - cos(ωT))/(k² + ω²). To minimize the average, we need to minimize this expression.Let me denote S = [k sin(ωT)]/(ω(k² + ω²)) + (1 - cos(ωT))/(k² + ω²)We need to minimize S with respect to ω.Let me write S as:S = [k sin(ωT)]/(ω(k² + ω²)) + (1 - cos(ωT))/(k² + ω²)To find the minimum, take derivative dS/dω and set to zero.Compute dS/dω:First term: d/dω [k sin(ωT)/(ω(k² + ω²))]Let me denote f(ω) = sin(ωT), g(ω) = ω(k² + ω²)Then, derivative is [k (T cos(ωT) * g(ω) - f(ω) * g’(ω)) ] / [g(ω)]²g’(ω) = (k² + ω²) + ω*(2ω) = k² + 3ω²So,= k [ T cos(ωT) * ω(k² + ω²) - sin(ωT) * (k² + 3ω²) ] / [ω²(k² + ω²)^2 ]Second term: d/dω [ (1 - cos(ωT))/(k² + ω²) ]= [ T sin(ωT) * (k² + ω²) - (1 - cos(ωT)) * 2ω ] / (k² + ω²)^2So, total derivative:dS/dω = [k (T ω(k² + ω²) cos(ωT) - (k² + 3ω²) sin(ωT)) ] / [ω²(k² + ω²)^2 ] + [ T sin(ωT) (k² + ω²) - 2ω(1 - cos(ωT)) ] / (k² + ω²)^2Set this equal to zero:[k (T ω(k² + ω²) cos(ωT) - (k² + 3ω²) sin(ωT)) ] / [ω²(k² + ω²)^2 ] + [ T sin(ωT) (k² + ω²) - 2ω(1 - cos(ωT)) ] / (k² + ω²)^2 = 0Multiply both sides by (k² + ω²)^2 to eliminate denominators:[k (T ω(k² + ω²) cos(ωT) - (k² + 3ω²) sin(ωT)) ] / ω² + [ T sin(ωT) (k² + ω²) - 2ω(1 - cos(ωT)) ] = 0This is still very complicated. Maybe we can factor out some terms or look for a pattern.Alternatively, perhaps we can consider specific values or make approximations. For example, if ω is small, then terms with ω² can be neglected compared to k². Let's see:If ω << k, then k² + ω² ≈ k², and terms like ω² can be neglected.So, approximate the equation:First term:[k (T ω k² cos(ωT) - k² sin(ωT)) ] / ω² + [ T sin(ωT) k² - 2ω(1 - cos(ωT)) ] ≈ 0Simplify:= [k (k² T ω cos(ωT) - k² sin(ωT)) ] / ω² + [ k² T sin(ωT) - 2ω(1 - cos(ωT)) ]= [k³ (T ω cos(ωT) - sin(ωT)) ] / ω² + k² T sin(ωT) - 2ω(1 - cos(ωT))This still seems complicated, but maybe for small ω, we can approximate cos(ωT) ≈ 1 - (ωT)^2/2 and sin(ωT) ≈ ωT - (ωT)^3/6.But this might not lead us anywhere quickly.Alternatively, perhaps the optimal ω is such that the derivative of the steady-state term is zero, ignoring the transient term. But I'm not sure.Alternatively, maybe the average crime rate is minimized when the frequency ω is such that the policy interventions are most effective, which might be when ω is equal to k, but I'm not sure.Alternatively, perhaps we can consider that the average of the steady-state solution over a period is zero, but since T is fixed, it's not necessarily a period. So, the average would depend on the phase and frequency.Alternatively, maybe the average is minimized when the amplitude of the steady-state solution is minimized, which occurs when ω is as large as possible, but that might not be practical.Alternatively, perhaps the optimal ω is such that the derivative of the average with respect to ω is zero, which would require solving the complicated equation we derived earlier. But without further simplification or numerical methods, it's hard to find an explicit solution.Given the complexity, perhaps the answer involves setting the derivative expression to zero and solving for ω, but it's not straightforward. Alternatively, maybe the optimal ω is such that ωT = nπ for some integer n, but that might not necessarily minimize the average.Alternatively, perhaps the average is minimized when the transient term is as small as possible, which would occur when ω is such that the steady-state term is minimized. But again, this is speculative.Given the time I've spent, maybe I should conclude that the optimal ω is such that the derivative expression equals zero, which would require solving the equation:[k (T ω(k² + ω²) cos(ωT) - (k² + 3ω²) sin(ωT)) ] / [ω²(k² + ω²)^2 ] + [ T sin(ωT) (k² + ω²) - 2ω(1 - cos(ωT)) ] / (k² + ω²)^2 = 0But this is too complicated to solve analytically, so perhaps the answer is that the optimal ω satisfies this equation, which would need to be solved numerically.Alternatively, maybe there's a simpler approach. Let me think again.The average crime rate is:Average C = (1/T)[ (C0 - k/(k² + ω²))(1 - e^(-kT))/k + (k sin(ωT))/(ω(k² + ω²)) + (1 - cos(ωT))/(k² + ω²) ]To minimize this, we can consider that the terms involving ω are:Term A: -k/(k² + ω²)Term B: (k sin(ωT))/(ω(k² + ω²))Term C: (1 - cos(ωT))/(k² + ω²)So, the total expression is:Average C = (1/T)[ C0*(1 - e^(-kT))/k + Term A*(1 - e^(-kT))/k + Term B + Term C ]We need to minimize this with respect to ω.Alternatively, perhaps we can consider that the optimal ω is such that the derivative of the sum of Term B and Term C is zero.Let me denote F(ω) = Term B + Term C = (k sin(ωT))/(ω(k² + ω²)) + (1 - cos(ωT))/(k² + ω²)We need to find ω that minimizes F(ω).Taking derivative F’(ω):F’(ω) = [k (T ω(k² + ω²) cos(ωT) - (k² + 3ω²) sin(ωT)) ] / [ω²(k² + ω²)^2 ] + [ T sin(ωT) (k² + ω²) - 2ω(1 - cos(ωT)) ] / (k² + ω²)^2Set F’(ω) = 0.This is the same equation as before. So, perhaps the optimal ω is the solution to this equation.But without further simplification, I think this is as far as we can go analytically. Therefore, the values of ω that minimize the average crime rate are the solutions to the equation:[k (T ω(k² + ω²) cos(ωT) - (k² + 3ω²) sin(ωT)) ] / [ω²(k² + ω²)^2 ] + [ T sin(ωT) (k² + ω²) - 2ω(1 - cos(ωT)) ] / (k² + ω²)^2 = 0This would typically require numerical methods to solve for ω given specific values of A, k, and T.But perhaps there's a simpler way. Let me consider that for the average to be minimized, the oscillatory terms should cancel out as much as possible. Maybe when ωT is a multiple of π, but I'm not sure.Alternatively, perhaps the optimal ω is such that the frequency matches the natural frequency, but in this case, the natural frequency is related to k, but it's a decay rate, not an oscillation.Alternatively, maybe the optimal ω is zero, but that would mean no policy intervention, which doesn't make sense.Alternatively, perhaps the optimal ω is such that the derivative of the average with respect to ω is zero, which is the equation we derived.Given the time I've spent, I think I'll have to conclude that the optimal ω satisfies the equation above, which would need to be solved numerically.</think>

🧠:<think>Okay, so I have this problem where a union worker is analyzing the impact of proposed changes in labor laws and wages on his income. The union is pushing for a 20% increase in the hourly wage and a reduction in the workweek from 40 hours to 35 hours. They want to make sure that the worker's weekly income remains the same or higher. There are two parts to this problem. The first part is to express the condition for his weekly income to remain the same or increase under the new proposal and solve for the minimum hourly wage W' that satisfies this condition. The second part is about calculating the minimum number of overtime hours needed to achieve a weekly income that's at least 10% higher than his current income, considering that any hours beyond 35 will be paid at 1.5 times the new hourly wage.Let me start with the first part.1. Current Situation: - Current hourly wage: W dollars per hour. - Current workweek: 40 hours. - Therefore, current weekly income is 40 * W.Proposed Changes: - Hourly wage increases by 20%, so the new wage is W' = W + 0.20W = 1.20W. - Workweek is reduced to 35 hours. - So, the new weekly income without overtime would be 35 * W'.But the union wants the worker's weekly income to remain the same or higher. So, the new income should be greater than or equal to the current income.So, mathematically, the condition is:35 * W' ≥ 40 * WBut since W' is 1.20W, let me substitute that in:35 * 1.20W ≥ 40WLet me compute 35 * 1.20:35 * 1.20 = 42So, 42W ≥ 40WSubtract 40W from both sides:2W ≥ 0Which is always true because W is a positive number (hourly wage can't be negative). Hmm, that seems odd. So, does this mean that the worker's income will automatically be higher with the new proposal?Wait, let me double-check my calculations.Current income: 40WProposed income: 35 * 1.20W = 42WSo, 42W is indeed greater than 40W, which means the worker's income increases by 2W. So, the condition is automatically satisfied because 42W is more than 40W. Therefore, the worker's income will increase under the new proposal without any need for overtime.But the question says "express the condition for his weekly income to remain the same or increase under the new proposal." So, maybe I need to express it in terms of W' without substituting the 20% increase yet.Wait, perhaps I misinterpreted the problem. Maybe the 20% increase is a proposal, but the worker wants to ensure that even if the workweek is reduced, his income doesn't drop. So, maybe the 20% increase is part of the proposal, but the worker wants to find the minimum W' such that 35W' is at least 40W.Wait, but if W' is 1.20W, then 35*1.20W is 42W, which is higher than 40W. So, perhaps the minimum W' is 1.20W, but that already satisfies the condition.Wait, maybe the problem is that the 20% increase is not guaranteed, and the union is advocating for it, but the worker wants to find the minimum W' such that 35W' ≥ 40W. So, solving for W':35W' ≥ 40WDivide both sides by 35:W' ≥ (40/35)WSimplify 40/35: that's 8/7 ≈ 1.142857So, W' must be at least approximately 14.2857% higher than the current wage.But the union is advocating for a 20% increase, which is higher than 14.2857%, so that would satisfy the condition.Wait, so maybe the first part is to express the condition as 35W' ≥ 40W, and solving for W' gives W' ≥ (40/35)W = (8/7)W ≈ 1.142857W.So, the minimum W' is 8/7 W or approximately 14.29% increase.But the union is proposing a 20% increase, which is more than that, so it's sufficient.So, perhaps the answer is W' ≥ (8/7)W, so the minimum W' is 8/7 W.Let me write that down.Problem 1 Answer:The condition is 35W' ≥ 40W, which simplifies to W' ≥ (8/7)W. Therefore, the minimum hourly wage W' is 8/7 times the current wage W.Now, moving on to the second part.2. The worker is considering the potential for overtime pay if the workweek is reduced. Suppose that any hours worked beyond 35 hours in a week will be paid at 1.5 times the new hourly wage W'. Calculate the minimum number of overtime hours the worker must work to achieve a weekly income that is at least 10% higher than his current weekly income.First, let's understand the current weekly income and the target income.Current Weekly Income:40WTarget Weekly Income:At least 10% higher than current, so:1.10 * 40W = 44WUnder the new proposal, the worker works 35 hours at W' per hour, and any hours beyond that are paid at 1.5W'.Let x be the number of overtime hours worked.So, the total weekly income with overtime is:35W' + x*(1.5W')We need this to be at least 44W.So, the inequality is:35W' + 1.5xW' ≥ 44WWe can factor out W':W'(35 + 1.5x) ≥ 44WBut from part 1, we know that W' is at least (8/7)W. However, in this case, the union is advocating for a 20% increase, so W' = 1.20W.Wait, the problem says "the new hourly wage W'", so I think we can assume that the 20% increase is already in place, so W' = 1.20W.So, substituting W' = 1.20W into the inequality:1.20W*(35 + 1.5x) ≥ 44WWe can divide both sides by W (since W > 0):1.20*(35 + 1.5x) ≥ 44Now, let's compute 1.20*35:1.20*35 = 42So, the inequality becomes:42 + 1.80x ≥ 44Subtract 42 from both sides:1.80x ≥ 2Now, solve for x:x ≥ 2 / 1.80Calculate 2 / 1.80:2 ÷ 1.8 = 1.111...So, x ≥ approximately 1.111 hours.Since the worker can't work a fraction of an hour in practice, but the problem doesn't specify rounding, so we can express it as 1.111... hours, which is 1 and 1/9 hours, or approximately 1.11 hours.But let me check my steps again.Starting from:35W' + 1.5xW' ≥ 44WWith W' = 1.20W:35*1.20W + 1.5x*1.20W ≥ 44WWhich is:42W + 1.80xW ≥ 44WSubtract 42W:1.80xW ≥ 2WDivide both sides by W (W > 0):1.80x ≥ 2x ≥ 2 / 1.80 = 10/9 ≈ 1.111...So, yes, that's correct.Therefore, the minimum number of overtime hours needed is 10/9 hours, which is approximately 1.11 hours.But let me think about this. If the worker works 1.11 hours overtime, that's about 1 hour and 6.66 minutes. Since in practice, overtime is usually counted in whole hours or fractions thereof, depending on the employer's policy. But the problem doesn't specify, so we can just present the exact value.Alternatively, if we need to express it as a fraction, 10/9 hours is the exact value.So, the minimum number of overtime hours is 10/9 hours.Let me write that down.Problem 2 Answer:The minimum number of overtime hours needed is 10/9 hours, which is approximately 1.11 hours.Wait, but let me double-check the calculations once more to be sure.Starting with the target income: 1.10 * 40W = 44W.Under the new proposal, without overtime, the worker earns 35W' = 35*1.20W = 42W.So, the difference between target and current is 44W - 42W = 2W.Each overtime hour gives 1.5W' = 1.5*1.20W = 1.80W.So, the number of overtime hours needed is 2W / 1.80W = 2/1.80 = 10/9 ≈ 1.111...Yes, that's correct.So, to achieve a 10% increase in income, the worker needs to work an additional 10/9 hours of overtime.Therefore, the answers are:1. The minimum W' is 8/7 W.2. The minimum overtime hours is 10/9 hours.</think>