🧠:<think>Okay, so I need to help analyze José Luis Friaza's soccer performance using this function ( f(t) = e^{-alpha t} cos(beta t) + gamma t ). The problem has two parts: first, finding the times ( t_1 ) and ( t_2 ) when his effectiveness peaks, and second, calculating the total effectiveness by integrating ( f(t) ) over 90 minutes.Starting with part 1: finding the peak times. Since the effectiveness score peaks at two different times, I know that the function ( f(t) ) must have two critical points where its derivative is zero. So, I need to find the derivative of ( f(t) ) with respect to ( t ) and set it equal to zero.Let me compute the derivative ( f'(t) ):( f(t) = e^{-alpha t} cos(beta t) + gamma t )So, the derivative is:( f'(t) = frac{d}{dt} [e^{-alpha t} cos(beta t)] + frac{d}{dt} [gamma t] )I can use the product rule for the first term. The derivative of ( e^{-alpha t} ) is ( -alpha e^{-alpha t} ), and the derivative of ( cos(beta t) ) is ( -beta sin(beta t) ). So,( frac{d}{dt} [e^{-alpha t} cos(beta t)] = -alpha e^{-alpha t} cos(beta t) - beta e^{-alpha t} sin(beta t) )And the derivative of ( gamma t ) is just ( gamma ).Putting it all together:( f'(t) = -alpha e^{-alpha t} cos(beta t) - beta e^{-alpha t} sin(beta t) + gamma )To find the critical points, set ( f'(t) = 0 ):( -alpha e^{-alpha t} cos(beta t) - beta e^{-alpha t} sin(beta t) + gamma = 0 )Let me factor out ( e^{-alpha t} ):( e^{-alpha t} (-alpha cos(beta t) - beta sin(beta t)) + gamma = 0 )So,( e^{-alpha t} (-alpha cos(beta t) - beta sin(beta t)) = -gamma )Multiply both sides by -1:( e^{-alpha t} (alpha cos(beta t) + beta sin(beta t)) = gamma )Hmm, this equation looks a bit complicated. Maybe I can write the expression inside the parentheses as a single sine or cosine function. I remember that expressions of the form ( A cos(theta) + B sin(theta) ) can be written as ( C cos(theta - phi) ) where ( C = sqrt{A^2 + B^2} ) and ( phi = arctan(B/A) ).Let me apply that here. Let ( A = alpha ) and ( B = beta ). Then,( alpha cos(beta t) + beta sin(beta t) = C cos(beta t - phi) ), where ( C = sqrt{alpha^2 + beta^2} ) and ( phi = arctan(beta / alpha) ).So, substituting back into the equation:( e^{-alpha t} C cos(beta t - phi) = gamma )Which simplifies to:( e^{-alpha t} cos(beta t - phi) = gamma / C )Let me denote ( D = gamma / C ), so:( e^{-alpha t} cos(beta t - phi) = D )This equation is still transcendental, meaning it can't be solved algebraically for ( t ). So, I might need to use some approximation or express the solutions in terms of inverse functions.But the problem says that there are two peak times ( t_1 ) and ( t_2 ) within 0 to 90 minutes. So, perhaps we can write the equations for ( t_1 ) and ( t_2 ) as solutions to:( e^{-alpha t} cos(beta t - phi) = D )But since this is transcendental, maybe I can express ( t ) in terms of the inverse function. Alternatively, perhaps the problem expects me to set up the equations without solving them explicitly.Wait, the question says "derive the equations for ( t_1 ) and ( t_2 ) in terms of ( alpha ), ( beta ), and ( gamma )." So, maybe it's sufficient to write the equation that ( t_1 ) and ( t_2 ) satisfy, which is the derivative set to zero.So, from earlier, we have:( e^{-alpha t} (alpha cos(beta t) + beta sin(beta t)) = gamma )So, the equations for ( t_1 ) and ( t_2 ) are:( e^{-alpha t_1} (alpha cos(beta t_1) + beta sin(beta t_1)) = gamma )and( e^{-alpha t_2} (alpha cos(beta t_2) + beta sin(beta t_2)) = gamma )Alternatively, using the amplitude form:( e^{-alpha t} sqrt{alpha^2 + beta^2} cos(beta t - phi) = gamma )So,( e^{-alpha t} cos(beta t - phi) = gamma / sqrt{alpha^2 + beta^2} )Let me denote ( K = gamma / sqrt{alpha^2 + beta^2} ), so:( e^{-alpha t} cos(beta t - phi) = K )So, the equations are:( e^{-alpha t_1} cos(beta t_1 - phi) = K )and( e^{-alpha t_2} cos(beta t_2 - phi) = K )But since ( phi = arctan(beta / alpha) ), we can write:( phi = arctan(beta / alpha) )So, maybe it's better to express the equations as:( e^{-alpha t} cos(beta t - arctan(beta / alpha)) = gamma / sqrt{alpha^2 + beta^2} )But perhaps the problem expects the equations in terms of sine and cosine without combining them. So, going back to the original derivative equation:( -alpha e^{-alpha t} cos(beta t) - beta e^{-alpha t} sin(beta t) + gamma = 0 )Which can be written as:( alpha e^{-alpha t} cos(beta t) + beta e^{-alpha t} sin(beta t) = gamma )So, factoring out ( e^{-alpha t} ):( e^{-alpha t} (alpha cos(beta t) + beta sin(beta t)) = gamma )Therefore, the equations for ( t_1 ) and ( t_2 ) are:( e^{-alpha t_1} (alpha cos(beta t_1) + beta sin(beta t_1)) = gamma )and( e^{-alpha t_2} (alpha cos(beta t_2) + beta sin(beta t_2)) = gamma )So, that's probably the answer for part 1.Moving on to part 2: calculating the total effectiveness score by integrating ( f(t) ) from 0 to 90.So, the integral is:( int_{0}^{90} f(t) dt = int_{0}^{90} e^{-alpha t} cos(beta t) dt + int_{0}^{90} gamma t dt )I can split the integral into two parts. The second integral is straightforward:( int_{0}^{90} gamma t dt = gamma int_{0}^{90} t dt = gamma left[ frac{t^2}{2} right]_0^{90} = gamma left( frac{90^2}{2} - 0 right) = gamma times 4050 = 4050 gamma )Now, the first integral is ( int e^{-alpha t} cos(beta t) dt ). I remember that this integral can be solved using integration by parts or by using a standard formula.The standard integral formula is:( int e^{at} cos(bt) dt = frac{e^{at}}{a^2 + b^2} (a cos(bt) + b sin(bt)) ) + C )But in our case, the exponent is negative, so ( a = -alpha ). Let me apply the formula accordingly.Let me denote ( a = -alpha ), ( b = beta ). Then,( int e^{a t} cos(b t) dt = frac{e^{a t}}{a^2 + b^2} (a cos(b t) + b sin(b t)) ) + C )So, substituting ( a = -alpha ) and ( b = beta ):( int e^{-alpha t} cos(beta t) dt = frac{e^{-alpha t}}{(-alpha)^2 + beta^2} (-alpha cos(beta t) + beta sin(beta t)) ) + C )Simplify the denominator:( (-alpha)^2 + beta^2 = alpha^2 + beta^2 )So,( int e^{-alpha t} cos(beta t) dt = frac{e^{-alpha t}}{alpha^2 + beta^2} (-alpha cos(beta t) + beta sin(beta t)) ) + C )Now, we need to evaluate this from 0 to 90.So, the definite integral is:( left[ frac{e^{-alpha t}}{alpha^2 + beta^2} (-alpha cos(beta t) + beta sin(beta t)) right]_0^{90} )Let me compute this at ( t = 90 ) and ( t = 0 ).At ( t = 90 ):( frac{e^{-90 alpha}}{alpha^2 + beta^2} (-alpha cos(90 beta) + beta sin(90 beta)) )At ( t = 0 ):( frac{e^{0}}{alpha^2 + beta^2} (-alpha cos(0) + beta sin(0)) = frac{1}{alpha^2 + beta^2} (-alpha times 1 + beta times 0) = frac{-alpha}{alpha^2 + beta^2} )So, the definite integral is:( frac{e^{-90 alpha}}{alpha^2 + beta^2} (-alpha cos(90 beta) + beta sin(90 beta)) - left( frac{-alpha}{alpha^2 + beta^2} right) )Simplify:( frac{e^{-90 alpha}}{alpha^2 + beta^2} (-alpha cos(90 beta) + beta sin(90 beta)) + frac{alpha}{alpha^2 + beta^2} )Factor out ( frac{1}{alpha^2 + beta^2} ):( frac{1}{alpha^2 + beta^2} left( e^{-90 alpha} (-alpha cos(90 beta) + beta sin(90 beta)) + alpha right) )So, putting it all together, the total effectiveness score is:( int_{0}^{90} f(t) dt = frac{1}{alpha^2 + beta^2} left( e^{-90 alpha} (-alpha cos(90 beta) + beta sin(90 beta)) + alpha right) + 4050 gamma )I think that's the expression for the total effectiveness.Let me just double-check my steps:1. For part 1, I found the derivative, set it to zero, and expressed the equations for ( t_1 ) and ( t_2 ). That seems correct.2. For part 2, I split the integral into two parts. The second integral was straightforward. The first integral, I used the standard formula, substituted the limits correctly, and simplified. It looks correct.So, I think these are the answers.</think>

🧠:<think>Okay, so I have this problem where I need to analyze the financial health of a small business in New Mexico over the next five years. They're planning to invest in two main areas: renewable energy installations and community development projects. I need to figure out the total energy savings from the renewable installations and determine if the business can sustain the community development investments based on their revenue growth.Starting with the renewable energy installations. The initial cost is 100,000. They expect these installations to reduce energy costs by 5% per year, compounded annually. The current annual energy cost is 50,000. Without the installations, energy costs would increase by 3% each year. I need to calculate the total energy savings over the next five years.Hmm, let's break this down. First, without the installations, the energy costs would increase by 3% each year. So, each year, the cost would be the previous year's cost multiplied by 1.03. With the installations, the energy costs would decrease by 5% each year, so each year's cost would be the previous year's cost multiplied by 0.95.But wait, the initial energy cost is 50,000. So, without the installations, the cost each year would be:Year 1: 50,000 * 1.03Year 2: (50,000 * 1.03) * 1.03 = 50,000 * (1.03)^2And so on up to Year 5.Similarly, with the installations, the energy cost each year would be:Year 1: 50,000 * 0.95Year 2: (50,000 * 0.95) * 0.95 = 50,000 * (0.95)^2Up to Year 5.But actually, the installations start reducing the cost from the first year. So, the savings each year would be the difference between the cost without installations and the cost with installations.So, for each year from 1 to 5, I need to calculate both the cost without and with the installations, subtract them to get the savings, and then sum all those savings.Alternatively, maybe there's a formula for the present value or future value of these savings, but since we're just summing the savings over five years, it might be straightforward to calculate each year's savings and add them up.Let me try that approach.First, calculate the energy costs without installations for each year:Year 1: 50,000 * 1.03 = 51,500Year 2: 51,500 * 1.03 = 53,045Year 3: 53,045 * 1.03 ≈ 54,636.35Year 4: 54,636.35 * 1.03 ≈ 56,275.44Year 5: 56,275.44 * 1.03 ≈ 58,003.69Now, with the installations, the energy costs each year:Year 1: 50,000 * 0.95 = 47,500Year 2: 47,500 * 0.95 = 45,125Year 3: 45,125 * 0.95 ≈ 42,868.75Year 4: 42,868.75 * 0.95 ≈ 40,725.31Year 5: 40,725.31 * 0.95 ≈ 38,689.05Now, the savings each year are the difference between the two:Year 1: 51,500 - 47,500 = 4,000Year 2: 53,045 - 45,125 = 7,920Year 3: 54,636.35 - 42,868.75 ≈ 11,767.60Year 4: 56,275.44 - 40,725.31 ≈ 15,550.13Year 5: 58,003.69 - 38,689.05 ≈ 19,314.64Now, summing these savings:4,000 + 7,920 = 11,92011,920 + 11,767.60 ≈ 23,687.6023,687.60 + 15,550.13 ≈ 39,237.7339,237.73 + 19,314.64 ≈ 58,552.37So, the total energy savings over five years would be approximately 58,552.37.But wait, is this correct? Because the initial cost of the installations is 100,000. So, the net savings would be the total savings minus the initial investment. But the question says "Calculate the total energy savings over the next 5 years," so I think it's just the sum of the savings each year, not considering the initial cost. So, 58,552.37 is the total energy savings.Alternatively, maybe we need to consider the time value of money, but the problem doesn't specify discounting, so I think it's just the nominal savings.Now, moving on to the community development projects. The business plans to invest 20,000 per year, expecting an ROI of 8% per year, compounded annually. However, they can only sustain this investment if their annual revenue increases by at least 4% per year. The current annual revenue is 200,000. We need to determine if the business can sustain this investment over five years.So, the business is investing 20,000 each year, which will grow at 8% annually. The return from these investments will be the total amount after five years. But the business needs to ensure that their revenue grows enough to cover these investments.Wait, actually, the ROI is 8% per year. So, each year's investment of 20,000 will earn 8% annually. So, the total return after five years would be the sum of each year's investment growing for the remaining years.Alternatively, the total return can be calculated as the future value of an annuity. The formula for the future value of an ordinary annuity is:FV = P * [(1 + r)^n - 1] / rWhere P is the annual payment, r is the interest rate, and n is the number of periods.So, plugging in the numbers:P = 20,000r = 0.08n = 5FV = 20,000 * [(1.08)^5 - 1] / 0.08First, calculate (1.08)^5:1.08^1 = 1.081.08^2 = 1.16641.08^3 ≈ 1.2597121.08^4 ≈ 1.360488961.08^5 ≈ 1.4693280768So, (1.4693280768 - 1) = 0.4693280768Divide by 0.08: 0.4693280768 / 0.08 ≈ 5.86660096Multiply by 20,000: 20,000 * 5.86660096 ≈ 117,332.02So, the future value of the community development investments after five years is approximately 117,332.02.But the business needs to sustain this investment, meaning they need to have enough revenue to cover the 20,000 annual investments. The revenue needs to increase by at least 4% per year.Current revenue is 200,000. Let's calculate the revenue each year with 4% growth:Year 1: 200,000 * 1.04 = 208,000Year 2: 208,000 * 1.04 = 216,320Year 3: 216,320 * 1.04 ≈ 225,004.80Year 4: 225,004.80 * 1.04 ≈ 233,854.99Year 5: 233,854.99 * 1.04 ≈ 243,099.19So, the revenue each year is increasing, but the business is investing 20,000 each year. The question is whether the revenue growth is sufficient to sustain these investments. I think this means that the revenue should be enough to cover the investments each year, but since the investments are being made from the revenue, we need to see if the revenue is growing enough to allow for the 20,000 investments without depleting the business.Alternatively, maybe the business needs to have enough revenue each year to cover the 20,000 investment. Since the revenue is growing, the 20,000 is a smaller proportion each year.But perhaps more accurately, the business's revenue needs to grow by 4% each year, so the required revenue each year is as calculated above. The business is investing 20,000 each year, which is part of their expenses or capital expenditures. The question is whether the revenue growth will allow them to sustain this investment without financial strain.Alternatively, maybe the business needs to ensure that the return from the community development projects (which is 117,332.02) is sufficient to cover the total investments made, which is 20,000 * 5 = 100,000. But since the ROI is 8%, the future value is higher than the total investment, so the business is making a profit on these investments.But the main point is whether the business can sustain the 20,000 annual investment given that their revenue needs to grow by at least 4% per year. Since the revenue is growing, the 20,000 is a smaller portion each year, so it should be sustainable.Wait, let's think differently. The business's revenue needs to increase by 4% each year to sustain the 20,000 investment. So, each year, the revenue must be at least 4% higher than the previous year to cover the investment. Since the revenue is growing by 4%, which is the required rate, the business can sustain the investment.But actually, the revenue is growing by 4%, so the required revenue each year is as calculated. The business is investing 20,000 each year, which is a fixed amount. Since the revenue is increasing, the proportion of revenue used for investment is decreasing each year, which is sustainable.Alternatively, if the revenue didn't grow, the business would have to allocate a larger portion of their revenue each year, which might not be sustainable. But since the revenue is growing by 4%, the 20,000 becomes a smaller burden each year.Therefore, the business can sustain the investment program over the next five years because their revenue is growing at the required rate of 4% per year.So, summarizing:1. Total energy savings over five years: approximately 58,552.372. The business can sustain the community development investments because their revenue is growing by 4% annually, which meets the required rate to sustain the 20,000 annual investments.</think>

🧠:<think>Okay, so I have this problem about a rockstar who created his own musical scale using a transformation function. The base note is 440 Hz, which I know is the standard tuning for an A note. But here, he's using a non-standard tuning system, so the frequencies aren't the usual ones we hear.The first part asks me to find the value of α, given that the 5th note in the scale is 880 Hz. The transformation function is given as f(n) = 440 × e^(αn). So, let me break this down.First, I know that n is the note number. So, the 5th note would be when n=5. The frequency at that point is 880 Hz. So, plugging into the formula:f(5) = 440 × e^(α × 5) = 880So, I can set up the equation:440 × e^(5α) = 880I need to solve for α. Let me divide both sides by 440 to simplify:e^(5α) = 880 / 440e^(5α) = 2Now, to solve for α, I need to take the natural logarithm of both sides. Remember that ln(e^x) = x.ln(e^(5α)) = ln(2)5α = ln(2)So, α = ln(2) / 5Let me compute that. I know ln(2) is approximately 0.6931, so:α ≈ 0.6931 / 5 ≈ 0.1386So, α is approximately 0.1386. But maybe I should keep it exact for now, so α = (ln 2)/5.Alright, that seems straightforward. Let me double-check my steps.1. f(n) = 440e^(αn)2. f(5) = 8803. 440e^(5α) = 8804. e^(5α) = 25. 5α = ln(2)6. α = (ln 2)/5 ≈ 0.1386Yep, that looks correct.Now, moving on to the second part. It asks for the interval ratio between the 3rd and 7th notes, which is f(7)/f(3). I need to compute this ratio using the α found in part 1.First, let me write expressions for f(7) and f(3).f(7) = 440 × e^(α × 7)f(3) = 440 × e^(α × 3)So, the ratio f(7)/f(3) would be:(440 × e^(7α)) / (440 × e^(3α)) = e^(7α - 3α) = e^(4α)Simplify that, it's e^(4α). Since we already know α = (ln 2)/5, let's plug that in.e^(4 × (ln 2)/5) = e^( (4/5) ln 2 )I can rewrite this using exponent rules. Remember that e^(a ln b) = b^a.So, e^( (4/5) ln 2 ) = 2^(4/5)Which is the 5th root of 2^4, or 2^(0.8). Let me compute that.2^(0.8) is approximately... Let me think. 2^1 = 2, 2^0.5 ≈ 1.414, 2^0.8 is somewhere between 1.414 and 2. Maybe around 1.741?But maybe I should just leave it as 2^(4/5) or write it as the 5th root of 16, since 2^4 is 16.Wait, 2^(4/5) is the same as the 5th root of 16, right? Because (16)^(1/5) = 2^(4/5). So, that's another way to express it.Alternatively, if I want to compute it numerically, I can use a calculator. Let me see:ln(2) ≈ 0.6931, so 4/5 of that is 0.6931 × 0.8 ≈ 0.5545So, e^(0.5545) ≈ e^0.5545. Let me compute e^0.5545.I know that e^0.5 ≈ 1.6487, and e^0.6 ≈ 1.8221. So, 0.5545 is between 0.5 and 0.6.Let me use linear approximation or maybe just remember that e^0.5545 is approximately 1.741.Alternatively, I can compute it more accurately.Let me use the Taylor series for e^x around x=0.5:e^x ≈ e^0.5 + e^0.5*(x - 0.5) + (e^0.5/2)*(x - 0.5)^2 + ...But maybe that's too complicated. Alternatively, I can use a calculator-like approach.We know that e^0.5545 ≈ ?Let me compute 0.5545:We know that ln(1.741) ≈ ?Wait, maybe I can reverse it. Let me compute ln(1.741):ln(1.741) ≈ 0.555, which is close to 0.5545. So, e^0.5545 ≈ 1.741.So, approximately 1.741.Therefore, the ratio f(7)/f(3) is approximately 1.741.But perhaps the exact form is better, so 2^(4/5). Alternatively, we can write it as 2^(0.8). But 2^(4/5) is more precise.Alternatively, since 2^(1/5) is the 5th root of 2, which is approximately 1.1487, so 2^(4/5) is (2^(1/5))^4 ≈ (1.1487)^4.Let me compute that:1.1487^2 ≈ 1.1487 × 1.1487 ≈ 1.3195Then, 1.3195^2 ≈ 1.3195 × 1.3195 ≈ 1.741Yes, so that's consistent. So, 2^(4/5) ≈ 1.741.So, the ratio is approximately 1.741.Alternatively, if I want to express it as a fraction, but since it's an irrational number, it's better to leave it in exponential form or as a decimal approximation.So, to recap:1. Found α = (ln 2)/5 ≈ 0.13862. Then, computed f(7)/f(3) = e^(4α) = 2^(4/5) ≈ 1.741I think that's the solution.Wait, let me double-check the ratio computation.f(7)/f(3) = e^(7α)/e^(3α) = e^(4α). Since α = (ln 2)/5, then 4α = (4 ln 2)/5, so e^(4α) = e^( (4/5) ln 2 ) = 2^(4/5). Yep, that's correct.Alternatively, if I compute 2^(4/5):2^(4/5) = (2^(1/5))^4.Since 2^(1/5) is approximately 1.1487, as I computed earlier, raising that to the 4th power gives approximately 1.741.So, that seems consistent.Alternatively, if I use logarithms:If I take the ratio f(7)/f(3) = 2^(4/5), then taking the logarithm base 2 of that ratio would give 4/5, which is 0.8, meaning it's an 80% increase in the exponent, which corresponds to a frequency ratio of 2^0.8.But in any case, the exact value is 2^(4/5), and the approximate decimal is 1.741.So, I think I've got both parts solved.Final Answer1. The value of ( alpha ) is boxed{dfrac{ln 2}{5}}.2. The interval ratio ( dfrac{f(7)}{f(3)} ) is boxed{2^{frac{4}{5}}}.</think>

🧠:<think>Okay, so I have this problem about a law student studying international trademark law. They're looking at how to allocate a marketing budget across five different countries to maximize the effectiveness of the trademark's global presence. Hmm, interesting. Let me try to break this down.First, the effectiveness in each country is given by the function ( E_i(x_i) = a_i ln(b_i x_i + 1) ). So, for each country ( i ), the effectiveness depends on the marketing budget allocated there, ( x_i ), and some constants ( a_i ) and ( b_i ). The total budget is ( B ), so the sum of all ( x_i ) should equal ( B ). The goal is to maximize the total effectiveness ( E = sum_{i=1}^{5} E_i(x_i) ).Alright, so part 1 is to formulate this as an optimization problem. I think this is a constrained optimization problem where we need to maximize the sum of these logarithmic functions subject to the total budget constraint. I remember that for such problems, we can use the method of Lagrange multipliers.Let me recall how Lagrange multipliers work. If we have a function to maximize, say ( f(x) ), subject to a constraint ( g(x) = 0 ), we introduce a multiplier ( lambda ) and set up the equation ( nabla f = lambda nabla g ). So, in this case, our function ( f ) is the total effectiveness ( E ), and the constraint ( g ) is ( sum x_i - B = 0 ).So, setting up the Lagrangian, it would be:( mathcal{L} = sum_{i=1}^{5} a_i ln(b_i x_i + 1) - lambda left( sum_{i=1}^{5} x_i - B right) )Then, to find the maximum, we take the partial derivatives of ( mathcal{L} ) with respect to each ( x_i ) and set them equal to zero.Let's compute the partial derivative for a general ( x_i ):( frac{partial mathcal{L}}{partial x_i} = frac{a_i b_i}{b_i x_i + 1} - lambda = 0 )So, for each country ( i ), we have:( frac{a_i b_i}{b_i x_i + 1} = lambda )This gives us a relationship between each ( x_i ) and the Lagrange multiplier ( lambda ). To solve for ( x_i ), we can rearrange the equation:( frac{a_i b_i}{lambda} = b_i x_i + 1 )Subtracting 1 from both sides:( frac{a_i b_i}{lambda} - 1 = b_i x_i )Dividing both sides by ( b_i ):( x_i = frac{a_i}{lambda} - frac{1}{b_i} )Hmm, so each ( x_i ) is expressed in terms of ( lambda ). But we also have the constraint that the sum of all ( x_i ) equals ( B ). So, let's sum up all the expressions for ( x_i ):( sum_{i=1}^{5} x_i = sum_{i=1}^{5} left( frac{a_i}{lambda} - frac{1}{b_i} right) = B )This simplifies to:( frac{sum_{i=1}^{5} a_i}{lambda} - sum_{i=1}^{5} frac{1}{b_i} = B )Let me denote ( A = sum_{i=1}^{5} a_i ) and ( C = sum_{i=1}^{5} frac{1}{b_i} ). Then, the equation becomes:( frac{A}{lambda} - C = B )Solving for ( lambda ):( frac{A}{lambda} = B + C )( lambda = frac{A}{B + C} )Now that we have ( lambda ), we can substitute back into the expression for each ( x_i ):( x_i = frac{a_i}{lambda} - frac{1}{b_i} = frac{a_i (B + C)}{A} - frac{1}{b_i} )So, each country's allocation ( x_i ) depends on its ( a_i ) and ( b_i ), as well as the total ( A ) and ( C ). This gives us the optimal allocation across the five countries.Moving on to part 2, the student finds that in country 3, the legal environment changes, doubling ( a_3 ). So, ( a_3 ) becomes ( 2a_3 ). We need to reformulate the optimization problem and explain how this affects the allocation strategy.First, let's see how the change affects the total ( A ) and ( C ). Originally, ( A = a_1 + a_2 + a_3 + a_4 + a_5 ). After doubling ( a_3 ), the new ( A' = a_1 + a_2 + 2a_3 + a_4 + a_5 = A + a_3 ).Similarly, ( C = sum_{i=1}^{5} frac{1}{b_i} ). Since only ( a_3 ) changes, ( C ) remains the same.So, the new Lagrange multiplier ( lambda' ) would be:( lambda' = frac{A'}{B + C} = frac{A + a_3}{B + C} )Comparing this to the original ( lambda = frac{A}{B + C} ), we see that ( lambda' ) is larger because ( A' > A ).Now, let's look at how each ( x_i ) changes. For country 3, the new allocation ( x_3' ) is:( x_3' = frac{2a_3}{lambda'} - frac{1}{b_3} )Substituting ( lambda' ):( x_3' = frac{2a_3 (B + C)}{A + a_3} - frac{1}{b_3} )Comparing this to the original ( x_3 = frac{a_3 (B + C)}{A} - frac{1}{b_3} ), we can see that ( x_3' ) is larger than ( x_3 ) because the numerator is doubled and the denominator is increased by ( a_3 ), but the effect of doubling might dominate.For the other countries, their allocations ( x_i' ) (where ( i neq 3 )) would be:( x_i' = frac{a_i}{lambda'} - frac{1}{b_i} = frac{a_i (B + C)}{A + a_3} - frac{1}{b_i} )Comparing to the original ( x_i = frac{a_i (B + C)}{A} - frac{1}{b_i} ), since ( A + a_3 > A ), each ( x_i' ) is less than ( x_i ) for ( i neq 3 ).So, the change in country 3's ( a_3 ) leads to an increase in the allocation to country 3 and a decrease in allocations to the other countries. This makes sense because country 3 has become more responsive due to the change in legal environment, so it's now more effective to allocate more budget there.I should also consider whether the total budget is still satisfied. The new allocations should still sum up to ( B ). Let me verify:Sum of new allocations:( x_3' + sum_{i neq 3} x_i' = left( frac{2a_3 (B + C)}{A + a_3} - frac{1}{b_3} right) + sum_{i neq 3} left( frac{a_i (B + C)}{A + a_3} - frac{1}{b_i} right) )Simplify:( frac{2a_3 (B + C)}{A + a_3} + sum_{i neq 3} frac{a_i (B + C)}{A + a_3} - left( frac{1}{b_3} + sum_{i neq 3} frac{1}{b_i} right) )The first two terms combine to:( frac{(2a_3 + sum_{i neq 3} a_i)(B + C)}{A + a_3} = frac{(A + a_3)(B + C)}{A + a_3} = B + C )The last term is ( C ), so the total becomes:( B + C - C = B )Which satisfies the budget constraint. So, the reallocation is valid.In summary, doubling ( a_3 ) increases the optimal allocation to country 3 and decreases allocations to other countries, reflecting the higher responsiveness of country 3's market.Final Answer1. The optimization problem is formulated as maximizing ( sum_{i=1}^{5} a_i ln(b_i x_i + 1) ) subject to ( sum_{i=1}^{5} x_i = B ). The optimal allocation for each country ( i ) is ( x_i = frac{a_i (B + C)}{A} - frac{1}{b_i} ), where ( A = sum_{i=1}^{5} a_i ) and ( C = sum_{i=1}^{5} frac{1}{b_i} ).2. After doubling ( a_3 ), the optimal allocation increases for country 3 and decreases for others. The new allocation for country 3 is ( boxed{x_3' = frac{2a_3 (B + C)}{A + a_3} - frac{1}{b_3}} ), and the allocations for other countries are ( boxed{x_i' = frac{a_i (B + C)}{A + a_3} - frac{1}{b_i}} ) for ( i neq 3 ).boxed{x_3' = frac{2a_3 (B + C)}{A + a_3} - frac{1}{b_3}} quad text{and} quad boxed{x_i' = frac{a_i (B + C)}{A + a_3} - frac{1}{b_i}} text{ for } i neq 3.</think>