🧠:<think>Alright, so I have two problems here related to Superman comics. Let me tackle them one by one.Starting with the first problem: It's about modeling the growth of a comic collection over time using a differential equation. The equation given is ( frac{dC}{dt} = k cdot C(t) cdot (1000 - C(t)) ). This looks like a logistic growth model, right? The general form of a logistic equation is ( frac{dC}{dt} = rC(1 - frac{C}{K}) ), where ( r ) is the growth rate and ( K ) is the carrying capacity. Comparing that to our equation, it seems like ( K = 1000 ) and ( r = k ). So, that makes sense.The initial condition is ( C(0) = 50 ). I need to find the general solution for ( C(t) ) and then determine ( C(t) ) when ( C(5) = 200 ). Hmm, okay, so first, let's solve the differential equation.The logistic equation is separable, so I can rewrite it as:( frac{dC}{C(1000 - C)} = k , dt )To integrate this, I should use partial fractions on the left side. Let me set up the partial fractions:( frac{1}{C(1000 - C)} = frac{A}{C} + frac{B}{1000 - C} )Multiplying both sides by ( C(1000 - C) ):( 1 = A(1000 - C) + B C )Expanding:( 1 = 1000A - A C + B C )Grouping like terms:( 1 = 1000A + (B - A)C )Since this must hold for all ( C ), the coefficients of like terms must be equal on both sides. So,For the constant term: ( 1000A = 1 ) => ( A = frac{1}{1000} )For the ( C ) term: ( B - A = 0 ) => ( B = A = frac{1}{1000} )So, the partial fractions decomposition is:( frac{1}{C(1000 - C)} = frac{1}{1000} left( frac{1}{C} + frac{1}{1000 - C} right) )Therefore, the integral becomes:( int left( frac{1}{1000C} + frac{1}{1000(1000 - C)} right) dC = int k , dt )Integrating both sides:Left side:( frac{1}{1000} ln |C| - frac{1}{1000} ln |1000 - C| + C_1 )Right side:( k t + C_2 )Combining constants:( frac{1}{1000} ln left| frac{C}{1000 - C} right| = k t + C )Where ( C ) is the constant of integration. Now, let's apply the initial condition ( C(0) = 50 ):( frac{1}{1000} ln left( frac{50}{1000 - 50} right) = 0 + C )Simplify:( frac{1}{1000} ln left( frac{50}{950} right) = C )( C = frac{1}{1000} ln left( frac{1}{19} right) )So, plugging back into the equation:( frac{1}{1000} ln left( frac{C}{1000 - C} right) = k t + frac{1}{1000} ln left( frac{1}{19} right) )Multiply both sides by 1000:( ln left( frac{C}{1000 - C} right) = 1000 k t + ln left( frac{1}{19} right) )Exponentiate both sides:( frac{C}{1000 - C} = e^{1000 k t} cdot frac{1}{19} )Let me denote ( e^{1000 k t} ) as ( e^{kt} ) raised to 1000, but maybe it's better to keep it as is for now.Let me solve for ( C ):Multiply both sides by ( 1000 - C ):( C = frac{1}{19} e^{1000 k t} (1000 - C) )Expand the right side:( C = frac{1000}{19} e^{1000 k t} - frac{1}{19} e^{1000 k t} C )Bring the ( C ) term to the left:( C + frac{1}{19} e^{1000 k t} C = frac{1000}{19} e^{1000 k t} )Factor out ( C ):( C left( 1 + frac{1}{19} e^{1000 k t} right) = frac{1000}{19} e^{1000 k t} )Solve for ( C ):( C = frac{ frac{1000}{19} e^{1000 k t} }{ 1 + frac{1}{19} e^{1000 k t} } )Simplify numerator and denominator:Multiply numerator and denominator by 19:( C = frac{1000 e^{1000 k t}}{19 + e^{1000 k t}} )So, that's the general solution.Now, we need to determine ( C(t) ) when ( C(5) = 200 ). Wait, actually, the problem says "determine ( C(t) ) when ( C(5) = 200 )". Hmm, does that mean we need to find the specific solution given that at ( t = 5 ), ( C = 200 )? So, we can use this to find the constant ( k ).So, let's plug ( t = 5 ) and ( C = 200 ) into the general solution:( 200 = frac{1000 e^{1000 k cdot 5}}{19 + e^{1000 k cdot 5}} )Let me denote ( e^{5000 k} = x ) for simplicity.So, equation becomes:( 200 = frac{1000 x}{19 + x} )Multiply both sides by ( 19 + x ):( 200(19 + x) = 1000 x )Expand:( 3800 + 200x = 1000x )Subtract 200x:( 3800 = 800x )Divide:( x = 3800 / 800 = 38 / 8 = 19 / 4 = 4.75 )So, ( x = e^{5000 k} = 4.75 )Take natural logarithm:( 5000 k = ln(4.75) )Calculate ( ln(4.75) ):I know that ( ln(4) approx 1.386 ), ( ln(5) approx 1.609 ). 4.75 is closer to 5, so maybe around 1.558? Let me compute it more accurately.Using calculator approximation:( ln(4.75) approx 1.5581 )So, ( k = frac{1.5581}{5000} approx 0.0003116 )So, ( k approx 0.0003116 ) per year.Therefore, the specific solution is:( C(t) = frac{1000 e^{1000 cdot 0.0003116 t}}{19 + e^{1000 cdot 0.0003116 t}} )Simplify exponent:1000 * 0.0003116 = 0.3116So,( C(t) = frac{1000 e^{0.3116 t}}{19 + e^{0.3116 t}} )Alternatively, we can write it as:( C(t) = frac{1000}{19 e^{-0.3116 t} + 1} )But perhaps it's better to leave it in the exponential form.So, that's the specific solution.Now, moving on to the second problem: The enthusiast is organizing a comic marathon, reading comics each day with a rate ( r(d) = a e^{b d} ), where ( d ) is the day number. They plan to finish reading their entire collection in exactly ( n ) days, and the total number of comics read is 1000. They read 10 comics on the first day and 20 on the last day.So, we need to find ( a ) and ( b ) given that ( r(1) = 10 ) and ( r(n) = 20 ), and the total sum ( sum_{d=1}^{n} r(d) = 1000 ).Wait, hold on. The rate function is given as ( r(d) = a e^{b d} ). So, each day, they read ( a e^{b d} ) comics. The total number of comics read is the sum from ( d = 1 ) to ( d = n ) of ( r(d) ), which is 1000.Given that on day 1, ( r(1) = 10 ), so:( a e^{b cdot 1} = 10 ) => ( a e^{b} = 10 ) ... (1)On day ( n ), ( r(n) = 20 ):( a e^{b n} = 20 ) ... (2)We have two equations:1. ( a e^{b} = 10 )2. ( a e^{b n} = 20 )We can divide equation (2) by equation (1):( frac{a e^{b n}}{a e^{b}} = frac{20}{10} )Simplify:( e^{b(n - 1)} = 2 )Take natural logarithm:( b(n - 1) = ln 2 )So,( b = frac{ln 2}{n - 1} )Now, from equation (1):( a = frac{10}{e^{b}} = 10 e^{-b} )So, ( a = 10 e^{- frac{ln 2}{n - 1}} = 10 cdot 2^{- frac{1}{n - 1}} )So, now, we have expressions for ( a ) and ( b ) in terms of ( n ). But we also know that the total number of comics read is 1000. So, the sum ( sum_{d=1}^{n} a e^{b d} = 1000 ).Let me write that sum:( sum_{d=1}^{n} a e^{b d} = a e^{b} sum_{d=0}^{n - 1} e^{b d} = 1000 )Wait, because ( a e^{b d} ) for ( d = 1 ) to ( n ) is the same as ( a e^{b} ) times the sum from ( d = 0 ) to ( n - 1 ) of ( e^{b d} ). That is, it's a geometric series.The sum ( sum_{d=0}^{n - 1} e^{b d} = frac{1 - e^{b n}}{1 - e^{b}} )So, plugging back in:( a e^{b} cdot frac{1 - e^{b n}}{1 - e^{b}} = 1000 )But from equation (1), ( a e^{b} = 10 ), and from equation (2), ( e^{b n} = frac{20}{a} ). Wait, actually, equation (2) is ( a e^{b n} = 20 ), so ( e^{b n} = frac{20}{a} ). But since ( a e^{b} = 10 ), ( a = 10 e^{-b} ), so ( e^{b n} = frac{20}{10 e^{-b}} = 2 e^{b} ).So, ( e^{b n} = 2 e^{b} )Therefore, ( e^{b(n - 1)} = 2 ), which is consistent with what we had earlier.So, going back to the sum:( 10 cdot frac{1 - e^{b n}}{1 - e^{b}} = 1000 )Simplify:( frac{1 - e^{b n}}{1 - e^{b}} = 100 )But we know ( e^{b n} = 2 e^{b} ), so substitute that in:( frac{1 - 2 e^{b}}{1 - e^{b}} = 100 )Let me denote ( x = e^{b} ). Then the equation becomes:( frac{1 - 2x}{1 - x} = 100 )Multiply both sides by ( 1 - x ):( 1 - 2x = 100(1 - x) )Expand:( 1 - 2x = 100 - 100x )Bring all terms to left:( 1 - 2x - 100 + 100x = 0 )Simplify:( -99 + 98x = 0 )So,( 98x = 99 )( x = frac{99}{98} approx 1.0102 )But ( x = e^{b} ), so:( e^{b} = frac{99}{98} )Take natural logarithm:( b = ln left( frac{99}{98} right) approx ln(1.0102) approx 0.01015 )So, ( b approx 0.01015 )But earlier, we had ( b = frac{ln 2}{n - 1} ). So,( frac{ln 2}{n - 1} approx 0.01015 )Solve for ( n ):( n - 1 = frac{ln 2}{0.01015} approx frac{0.6931}{0.01015} approx 68.28 )So, ( n approx 69.28 ). Since ( n ) must be an integer (number of days), we can round it to 69 days.But let's check if this makes sense. If ( n = 69 ), then ( b = frac{ln 2}{68} approx 0.01015 ), which matches our earlier calculation.Now, let's find ( a ):From equation (1):( a = 10 e^{-b} approx 10 e^{-0.01015} approx 10 times 0.9899 approx 9.899 )So, ( a approx 9.899 ), and ( b approx 0.01015 )But let's verify the total sum with these values.The sum is:( sum_{d=1}^{69} a e^{b d} = a e^{b} cdot frac{1 - e^{b cdot 69}}{1 - e^{b}} )We know ( a e^{b} = 10 ), and ( e^{b cdot 69} = e^{b cdot (68 + 1)} = e^{b cdot 68} cdot e^{b} ). From earlier, ( e^{b cdot 68} = 2 ), so ( e^{b cdot 69} = 2 e^{b} approx 2 times 1.0102 approx 2.0204 )Thus, the sum becomes:( 10 cdot frac{1 - 2.0204}{1 - 1.0102} = 10 cdot frac{-1.0204}{-0.0102} = 10 cdot 100.0392 approx 1000.392 )Which is approximately 1000, as required. So, our calculations are consistent.Therefore, the values are:( a approx 9.899 ) and ( b approx 0.01015 )But since the problem might expect exact expressions rather than approximate decimals, let's express them in terms of logarithms.We had:( b = frac{ln 2}{n - 1} )And ( a = 10 e^{-b} )But we found ( n approx 69 ), so ( n - 1 = 68 ). Therefore, ( b = frac{ln 2}{68} ), and ( a = 10 e^{- frac{ln 2}{68}} = 10 cdot 2^{- frac{1}{68}} )So, exact expressions are:( a = 10 cdot 2^{-1/68} )( b = frac{ln 2}{68} )Alternatively, we can write ( a = frac{10}{2^{1/68}} )So, that's the exact solution.Final Answer1. The specific solution for ( C(t) ) is ( boxed{C(t) = dfrac{1000 e^{0.3116 t}}{19 + e^{0.3116 t}}} ).2. The values of ( a ) and ( b ) are ( boxed{a = 10 cdot 2^{-1/68}} ) and ( boxed{b = dfrac{ln 2}{68}} ).</think>

🧠:<think>Alright, so I've got this problem about a quarterback's rating and some game strategy optimization. Let me try to figure this out step by step. I'm a bit new to this, so I might need to take it slow.Starting with the first part: calculating the quarterback rating (QBR). The formula is given as:QBR = [(A + B + C + D)/6] × 100Where A, B, C, D are defined as:A = (Completions/Attempts - 0.3)/0.2B = (Yards/Attempts - 3)/4C = (Touchdowns/Attempts)/0.05D = 2.375 - (Interceptions/Attempts)/0.04Okay, so I need to compute each of these components A, B, C, D using the given statistics:- Completions = 350- Attempts = 500- Yards = 4000- Touchdowns = 30- Interceptions = 10Let me compute each one step by step.First, A:A = (Completions/Attempts - 0.3)/0.2So, Completions/Attempts is 350/500. Let me calculate that. 350 divided by 500 is 0.7. Then subtract 0.3: 0.7 - 0.3 = 0.4. Then divide by 0.2: 0.4 / 0.2 = 2. So A is 2.Next, B:B = (Yards/Attempts - 3)/4Yards/Attempts is 4000/500. That's 8. Subtract 3: 8 - 3 = 5. Divide by 4: 5 / 4 = 1.25. So B is 1.25.Moving on to C:C = (Touchdowns/Attempts)/0.05Touchdowns/Attempts is 30/500. Let me compute that. 30 divided by 500 is 0.06. Then divide by 0.05: 0.06 / 0.05 = 1.2. So C is 1.2.Now, D:D = 2.375 - (Interceptions/Attempts)/0.04Interceptions/Attempts is 10/500. That's 0.02. Then divide by 0.04: 0.02 / 0.04 = 0.5. So subtract that from 2.375: 2.375 - 0.5 = 1.875. So D is 1.875.Now, let's sum up A, B, C, D:A + B + C + D = 2 + 1.25 + 1.2 + 1.875Let me add them up:2 + 1.25 = 3.253.25 + 1.2 = 4.454.45 + 1.875 = 6.325So the sum is 6.325.Now, divide that by 6:6.325 / 6 ≈ 1.054166...Then multiply by 100 to get QBR:1.054166... × 100 ≈ 105.4166...So the QBR is approximately 105.42.Wait, does that make sense? Let me double-check my calculations because sometimes I might have messed up a step.Starting with A: 350/500 is 0.7, minus 0.3 is 0.4, divided by 0.2 is 2. That seems right.B: 4000/500 is 8, minus 3 is 5, divided by 4 is 1.25. Correct.C: 30/500 is 0.06, divided by 0.05 is 1.2. Correct.D: 10/500 is 0.02, divided by 0.04 is 0.5, subtracted from 2.375 gives 1.875. Correct.Sum: 2 + 1.25 is 3.25, plus 1.2 is 4.45, plus 1.875 is 6.325. Divided by 6 is about 1.054, multiplied by 100 is 105.4. So yes, that seems correct.So the QBR is approximately 105.4, which is above 100. So the answer to the first part is yes, it exceeds 100.Moving on to the second part: Game Strategy Optimization.The function given is P(t) = -2t² + 24t - 60, where P(t) is passing yards at time t (in minutes). The fan wants to find the time t at which the quarterback achieves maximum passing yards and the maximum yards.This is a quadratic function, and since the coefficient of t² is negative (-2), the parabola opens downward, so the vertex is the maximum point.To find the time t at which the maximum occurs, we can use calculus. The maximum occurs where the derivative of P(t) with respect to t is zero.So, let's compute the derivative P'(t):P(t) = -2t² + 24t - 60P'(t) = dP/dt = -4t + 24Set the derivative equal to zero to find critical points:-4t + 24 = 0Solving for t:-4t = -24t = (-24)/(-4) = 6So, t = 6 minutes is the critical point.To confirm it's a maximum, we can check the second derivative or analyze the function.Second derivative P''(t) = d²P/dt² = -4, which is negative, confirming that the function is concave down, so t=6 is indeed a maximum.Now, let's compute the maximum passing yards P(6):P(6) = -2*(6)^2 + 24*(6) - 60Compute each term:-2*(36) = -7224*6 = 144So, P(6) = -72 + 144 - 60Compute step by step:-72 + 144 = 7272 - 60 = 12So, P(6) = 12 yards.Wait, that seems low. Is that correct?Wait, let me double-check the calculation:P(6) = -2*(6)^2 + 24*6 - 60First, 6 squared is 36.Multiply by -2: -72.24*6 is 144.So, -72 + 144 is 72.72 - 60 is 12.Hmm, 12 yards. That seems quite low for maximum passing yards. Maybe the function is scaled differently or perhaps it's yards per minute? Or maybe it's a typo in the function.Wait, the function is P(t) = -2t² + 24t - 60. So, at t=6, it's 12 yards. Maybe it's yards per minute? Or perhaps the function is defined differently.Alternatively, maybe the units are different. Wait, the problem says P(t) represents passing yards at any time t. So, it's total yards at time t minutes.But 12 yards at t=6 seems low. Let me check the calculations again.Wait, perhaps I made a mistake in the calculation.Wait, P(6) = -2*(6)^2 + 24*(6) -60Compute each term:-2*(36) = -7224*6 = 144So, -72 + 144 = 7272 - 60 = 12Yes, that's correct. So, the maximum passing yards is 12 yards at t=6 minutes.But that seems really low. Maybe the function is in yards per minute? Or perhaps it's a different unit.Wait, the problem says P(t) represents the passing yards at any time t. So, it's total yards. So, 12 yards in 6 minutes? That seems low, but maybe it's correct.Alternatively, perhaps the function is meant to be in a different context, like yards per play or something else. But as per the problem statement, it's passing yards at any time t.Alternatively, maybe I misread the function. Let me check again.The function is P(t) = -2t² + 24t -60. So, it's a quadratic function. The vertex is at t=6, and the maximum value is 12 yards.Alternatively, maybe the function is in terms of minutes into the game, and the yards are cumulative. So, maybe at t=6 minutes, the total passing yards are 12 yards. That seems low, but perhaps it's correct.Alternatively, maybe the function is supposed to be in terms of something else, but as per the problem, it's passing yards.Alternatively, maybe the function is in yards per minute, but then the maximum would be 12 yards per minute, which is 720 yards per hour, which is still high but maybe possible.Wait, but the function is P(t) = -2t² +24t -60. So, at t=0, P(0) = -60 yards? That doesn't make sense because yards can't be negative. Hmm, that's odd.Wait, that suggests that at time t=0, the passing yards are negative, which is impossible. So, perhaps the function is defined for t in a certain range where P(t) is positive.Wait, let's see when P(t) is positive.Set P(t) = 0:-2t² +24t -60 = 0Multiply both sides by -1:2t² -24t +60 = 0Divide by 2:t² -12t +30 = 0Use quadratic formula:t = [12 ± sqrt(144 - 120)] / 2sqrt(24) is approximately 4.899So, t = [12 ± 4.899]/2So, t ≈ (12 + 4.899)/2 ≈ 16.899/2 ≈ 8.4495t ≈ (12 - 4.899)/2 ≈ 7.101/2 ≈ 3.5505So, the function is zero at approximately t=3.55 and t=8.45. So, between t=3.55 and t=8.45, P(t) is positive.So, the maximum at t=6 is 12 yards, which is within the positive range.So, perhaps the function is correct, and the maximum yards are 12 yards at t=6 minutes.Alternatively, maybe the function is supposed to be in a different unit or scaled differently, but as per the problem, it's passing yards.So, perhaps the answer is that the maximum occurs at t=6 minutes, and the maximum yards are 12 yards.Alternatively, maybe I made a mistake in interpreting the function. Let me check the function again.P(t) = -2t² +24t -60Yes, that's correct.Alternatively, maybe the function is in yards per minute, so the maximum rate is 12 yards per minute at t=6.But the problem says P(t) represents the passing yards at any time t, so it's total yards, not rate.So, at t=6 minutes, the total passing yards are 12 yards.That seems low, but perhaps it's correct.Alternatively, maybe the function is supposed to be in a different context, but as per the problem, that's the case.So, I think the answer is t=6 minutes, maximum yards=12.Wait, but let me think again. Maybe the function is in terms of something else. Alternatively, perhaps the function is in yards per play or something else, but the problem says passing yards at any time t.Alternatively, maybe the function is supposed to be in terms of something else, but I think I have to go with the given information.So, to sum up:1. The QBR is approximately 105.42, which exceeds 100.2. The maximum passing yards occur at t=6 minutes, with a maximum of 12 yards.Wait, but 12 yards seems really low. Maybe I made a mistake in the calculation.Wait, let me recalculate P(6):P(6) = -2*(6)^2 +24*6 -60Compute each term:6 squared is 36.-2*36 = -7224*6 = 144So, -72 + 144 = 7272 -60 = 12Yes, that's correct.Alternatively, maybe the function is supposed to be in yards per minute, so the maximum rate is 12 yards per minute, which would be 720 yards per hour, which is high but possible.But the problem says P(t) represents passing yards at any time t, so it's total yards, not rate.So, perhaps the answer is correct.Alternatively, maybe the function is supposed to be in a different unit, but I think I have to go with the given information.So, I think the answers are:1. QBR ≈ 105.42, which is above 100.2. Maximum at t=6 minutes, with 12 yards.But 12 yards seems low, but maybe it's correct.Alternatively, perhaps the function is supposed to be in yards per minute, but the problem says passing yards, so it's total yards.Alternatively, maybe the function is supposed to be in a different context, but I think I have to go with the given information.So, I think that's the answer.</think>

🧠:<think>Okay, so I have this problem about a foreign correspondent studying voting patterns in two regions, A and B, in Ghana. She's collected data over three election cycles, which means t goes from 0 to 2, right? Because each election cycle is a year apart, I think. So, t=0 is the first election, t=1 is the second, and t=2 is the third.The first part asks me to calculate the total voter turnout in both regions over these three election cycles by integrating the respective functions. The functions given are f(t) = 5000 + 300t² for region A and g(t) = 8000 + 2000sin(π/2 t) for region B.Hmm, okay, so I need to integrate f(t) from t=0 to t=2 and g(t) from t=0 to t=2. That will give me the total number of voters in each region over the three election cycles. Let me write that down.For region A, the integral of f(t) from 0 to 2 is ∫₀² (5000 + 300t²) dt. Similarly, for region B, it's ∫₀² (8000 + 2000sin(π/2 t)) dt.Let me compute the integral for region A first. The integral of 5000 is straightforward; it's 5000t. The integral of 300t² is 300*(t³/3) which simplifies to 100t³. So putting it together, the integral from 0 to 2 is [5000t + 100t³] evaluated from 0 to 2.Calculating at t=2: 5000*2 + 100*(2)³ = 10,000 + 100*8 = 10,000 + 800 = 10,800.At t=0, it's 0 + 0 = 0. So the total voter turnout for region A is 10,800.Now, moving on to region B. The integral of g(t) is ∫₀² (8000 + 2000sin(π/2 t)) dt. Let's break this down. The integral of 8000 is 8000t. The integral of 2000sin(π/2 t) is a bit trickier. I remember that the integral of sin(ax) dx is -(1/a)cos(ax) + C. So here, a is π/2, so the integral becomes 2000*(-2/π)cos(π/2 t) + C, which simplifies to (-4000/π)cos(π/2 t).So putting it all together, the integral from 0 to 2 is [8000t - (4000/π)cos(π/2 t)] evaluated from 0 to 2.Let's compute this at t=2 first. 8000*2 = 16,000. Then, cos(π/2 * 2) = cos(π) = -1. So the second term is -(4000/π)*(-1) = 4000/π.So at t=2, the value is 16,000 + 4000/π.Now, at t=0: 8000*0 = 0. cos(π/2 * 0) = cos(0) = 1. So the second term is -(4000/π)*1 = -4000/π.So the value at t=0 is 0 - 4000/π.Therefore, the total integral from 0 to 2 is (16,000 + 4000/π) - (-4000/π) = 16,000 + 4000/π + 4000/π = 16,000 + 8000/π.Let me compute that numerically. 8000 divided by π is approximately 8000 / 3.1416 ≈ 2546.479. So adding that to 16,000 gives 16,000 + 2546.479 ≈ 18,546.479. So approximately 18,546.48 voters in region B.Wait, but the problem says "over the three election cycles," so does that mean t=0, t=1, t=2? Or is it continuous over the interval from t=0 to t=2? Hmm, the functions are given as continuous functions, so integrating from 0 to 2 makes sense as the total over the three cycles.So, region A has 10,800 voters total, and region B has approximately 18,546.48 voters total.Wait, but 18,546.48 is a decimal. Since voters are people, we should probably round to the nearest whole number. So 18,546 voters.Okay, so that's part 1 done. Now, part 2 says that region A's influence is directly proportional to the square root of the number of voters, and region B's influence is directly proportional to the logarithm of the number of voters. We need to find the ratio of the total influence of region A to region B over these cycles.So, let me denote the influence of A as k_A * sqrt(N_A), where N_A is the total voters in A, and influence of B as k_B * ln(N_B), where N_B is the total voters in B. But since the influence is directly proportional, the constants of proportionality would be the same if we're comparing ratios. Wait, actually, the problem says "directly proportional," so we can write influence A = k * sqrt(N_A) and influence B = k * ln(N_B). So the ratio would be sqrt(N_A) / ln(N_B), since the constants k would cancel out.But wait, is that correct? Or does each region have its own constant? The problem says "directly proportional," so it might mean that influence A = k_A * sqrt(N_A) and influence B = k_B * ln(N_B). But unless we know the constants, we can't compute the exact ratio. However, the problem says "the ratio of the total influence," so maybe it's just the ratio of their influences, assuming the same proportionality constant? Or perhaps the proportionality constants are the same because it's the same correspondent measuring both.Wait, the problem says "region A's voter influence... is directly proportional to the square root... and region B's influence is directly proportional to the logarithm..." So, it's possible that each has its own constant, but since we don't have values for the constants, maybe we can assume they are the same? Or perhaps we can consider the ratio as (sqrt(N_A) / ln(N_B)) times (k_A / k_B). But since we don't have k_A and k_B, maybe the question is just asking for sqrt(N_A) / ln(N_B), treating the constants as 1.Alternatively, maybe the influence is proportional, so we can write influence A = sqrt(N_A) and influence B = ln(N_B), with the proportionality constants being 1 for simplicity. So the ratio would be sqrt(N_A) / ln(N_B).But let me check the problem statement again: "region A's voter influence... is directly proportional to the square root... and region B's influence... is directly proportional to the logarithm..." So, it's possible that the proportionality constants are different, but since we don't have their values, perhaps we can assume they are the same, or perhaps the ratio is just sqrt(N_A)/ln(N_B). Let me proceed with that assumption.So, N_A is 10,800 and N_B is approximately 18,546.48.First, compute sqrt(N_A): sqrt(10,800). Let me calculate that. 10,800 is 108 * 100, so sqrt(108)*sqrt(100) = 10*sqrt(108). sqrt(108) is sqrt(36*3) = 6*sqrt(3) ≈ 6*1.732 ≈ 10.392. So 10*10.392 ≈ 103.92.Alternatively, using calculator: sqrt(10800) ≈ 103.923.Now, ln(N_B): ln(18,546.48). Let me compute that. I know that ln(10,000) is about 9.2103, ln(20,000) is about 9.9035. Since 18,546 is between 10,000 and 20,000, closer to 20,000. Let me compute it more accurately.Alternatively, using natural logarithm properties: ln(18,546.48) = ln(1.854648 * 10^4) = ln(1.854648) + ln(10^4) ≈ ln(1.854648) + 9.2103.Compute ln(1.854648): I know that ln(1.8) ≈ 0.5878, ln(1.85) is a bit more. Let me use a calculator approximation.Alternatively, using Taylor series or known values. Alternatively, since I don't have a calculator here, I can estimate it. Let me recall that ln(2) ≈ 0.6931, ln(1.6) ≈ 0.4700, ln(1.7) ≈ 0.5306, ln(1.8) ≈ 0.5878, ln(1.9) ≈ 0.6419, ln(2) ≈ 0.6931.1.854648 is between 1.8 and 1.9. Let's see, 1.854648 - 1.8 = 0.054648. The difference between ln(1.8) and ln(1.9) is about 0.6419 - 0.5878 = 0.0541. So, 0.054648 is almost the full difference, so ln(1.854648) ≈ 0.5878 + 0.054648*(0.0541/0.054648) ≈ wait, maybe a linear approximation.Let me denote x = 1.854648. Let me take x = 1.8 + 0.054648. Let me approximate ln(1.8 + Δx) ≈ ln(1.8) + (Δx)/1.8. So, Δx = 0.054648, so (Δx)/1.8 ≈ 0.054648 / 1.8 ≈ 0.03036. So ln(1.854648) ≈ 0.5878 + 0.03036 ≈ 0.61816.So, ln(18,546.48) ≈ 0.61816 + 9.2103 ≈ 9.8285.So, influence A is approximately 103.923, influence B is approximately 9.8285.Therefore, the ratio of influence A to influence B is 103.923 / 9.8285 ≈ let's compute that.103.923 ÷ 9.8285 ≈ approximately 10.57.Wait, let me do that division more accurately.9.8285 * 10 = 98.285Subtract that from 103.923: 103.923 - 98.285 = 5.638Now, 5.638 / 9.8285 ≈ 0.573So total ratio is 10 + 0.573 ≈ 10.573.So approximately 10.57.But let me check with more precise calculations.Alternatively, using calculator steps:Compute 103.923 / 9.8285:Divide numerator and denominator by 9.8285:103.923 ÷ 9.8285 ≈ (103.923 / 9.8285) ≈ 10.57.Yes, so approximately 10.57.But let me see if I can get a more precise value.Compute 9.8285 * 10.57:9.8285 * 10 = 98.2859.8285 * 0.57 ≈ 9.8285 * 0.5 = 4.91425, 9.8285 * 0.07 ≈ 0.688, so total ≈ 4.91425 + 0.688 ≈ 5.60225So total is 98.285 + 5.60225 ≈ 103.887, which is very close to 103.923. So the ratio is approximately 10.57.Therefore, the ratio of total influence of region A to region B is approximately 10.57.But let me check if I did everything correctly.Wait, for region B, the total voters were approximately 18,546.48, so ln(18,546.48) ≈ 9.8285. And sqrt(10,800) ≈ 103.923. So 103.923 / 9.8285 ≈ 10.57.Alternatively, if I use more precise values:sqrt(10,800) = sqrt(100*108) = 10*sqrt(108). sqrt(108) is sqrt(36*3) = 6*sqrt(3) ≈ 6*1.73205 ≈ 10.3923. So 10*10.3923 ≈ 103.923.For ln(18,546.48), let me compute it more accurately.We can use the fact that ln(18,546.48) = ln(18,546.48). Let me use natural logarithm properties.We know that ln(18,546.48) = ln(1.854648 * 10^4) = ln(1.854648) + ln(10^4) = ln(1.854648) + 9.2103.Now, let's compute ln(1.854648) more accurately.We can use the Taylor series expansion of ln(x) around x=1. Let me recall that ln(1 + y) ≈ y - y²/2 + y³/3 - y⁴/4 + ... for |y| < 1.But 1.854648 is 1 + 0.854648, which is more than 1, so maybe it's better to use another expansion point.Alternatively, use the known value of ln(1.8) ≈ 0.587787, ln(1.85) ≈ 0.615186, ln(1.854648) is slightly higher than ln(1.85).Compute the difference between 1.854648 and 1.85: 0.004648.So, using linear approximation around x=1.85:ln(1.85 + Δx) ≈ ln(1.85) + (Δx)/1.85.So, Δx = 0.004648.Thus, ln(1.854648) ≈ ln(1.85) + 0.004648 / 1.85 ≈ 0.615186 + 0.002512 ≈ 0.6177.So, ln(1.854648) ≈ 0.6177.Therefore, ln(18,546.48) ≈ 0.6177 + 9.2103 ≈ 9.828.So, that's consistent with my earlier approximation.Therefore, influence A is 103.923, influence B is 9.828.So, the ratio is 103.923 / 9.828 ≈ 10.57.So, approximately 10.57.But let me see if I can express this as an exact fraction or something, but probably not necessary. So, the ratio is approximately 10.57, which can be written as 10.57:1.Alternatively, if we want to express it as a fraction, 103.923 / 9.828 ≈ 10.57, which is roughly 10.57/1.But perhaps we can write it as a fraction. Let me see:103.923 / 9.828 ≈ 10.57.But 10.57 is approximately 1057/100, but that's not helpful.Alternatively, perhaps we can keep it as a decimal, 10.57.Alternatively, if we use exact values:sqrt(10800) = sqrt(100*108) = 10*sqrt(108) = 10*sqrt(36*3) = 10*6*sqrt(3) = 60*sqrt(3).And ln(18546.48) is approximately 9.8285.So, the ratio is 60*sqrt(3) / 9.8285.But 60*sqrt(3) ≈ 60*1.732 ≈ 103.92.So, 103.92 / 9.8285 ≈ 10.57.Alternatively, if we want to express it in terms of sqrt(3), but I think the problem expects a numerical value.So, the ratio is approximately 10.57.But let me check if I made any mistakes in the integration.For region A: ∫₀² (5000 + 300t²) dt = [5000t + 100t³] from 0 to 2.At t=2: 5000*2 = 10,000; 100*(8) = 800; total 10,800. Correct.For region B: ∫₀² (8000 + 2000sin(π/2 t)) dt.Integral of 8000 is 8000t.Integral of 2000sin(π/2 t) is -2000*(2/π)cos(π/2 t) = -4000/π cos(π/2 t).So, evaluated from 0 to 2:At t=2: 8000*2 = 16,000; -4000/π cos(π) = -4000/π*(-1) = 4000/π.At t=0: 8000*0 = 0; -4000/π cos(0) = -4000/π*(1) = -4000/π.So, total integral is (16,000 + 4000/π) - (-4000/π) = 16,000 + 8000/π.Yes, that's correct.8000/π ≈ 2546.479, so total ≈ 16,000 + 2546.479 ≈ 18,546.479.So, that's correct.Therefore, the calculations are correct.So, the ratio is approximately 10.57.But let me see if I can write it as an exact value.Influence A: sqrt(10800) = 60*sqrt(3).Influence B: ln(18546.48). But 18546.48 is 8000 + 2000*(1) at t=2? Wait, no, the integral is 16,000 + 8000/π, which is approximately 18,546.48.But 16,000 + 8000/π is the exact value, so ln(16,000 + 8000/π).So, the exact ratio is (60√3) / ln(16,000 + 8000/π).But that's a bit messy, so probably better to leave it as a decimal approximation.So, the ratio is approximately 10.57.Therefore, the final answer is approximately 10.57.But let me check if I can write it as a fraction. 10.57 is approximately 10 and 57/100, which is 1057/100, but that's not a simplified fraction. Alternatively, 10.57 is approximately 10.57, so we can write it as 10.57:1.Alternatively, if we want to express it as a ratio, it's about 10.57 to 1.So, I think that's the answer.Final Answer1. The total voter turnout in region A is boxed{10800} and in region B is approximately boxed{18546}.2. The ratio of the total influence of region A to region B is approximately boxed{10.57}.</think>

🧠:<think>Alright, so I need to figure out the total length of Captain John's flight path and the total variation in altitude. Let's start with the first part: calculating the total length. The flight path is made up of three segments, each defined by parametric equations. I remember that the length of a parametric curve can be found by integrating the magnitude of its derivative over the parameter interval. So, for each segment, I'll find the derivative, compute its magnitude, and then integrate that from 0 to 1. After that, I'll add up the lengths of all three segments to get the total flight path length.Starting with Segment 1: ( mathbf{r}_1(t) = (2t, 3t, 5t) ) for ( t ) in [0, 1]. First, I'll find the derivative of ( mathbf{r}_1(t) ) with respect to t. The derivative, ( mathbf{r}_1'(t) ), is (2, 3, 5). Next, I need the magnitude of this derivative. The magnitude is calculated as the square root of the sum of the squares of the components. So that would be ( sqrt{2^2 + 3^2 + 5^2} = sqrt{4 + 9 + 25} = sqrt{38} ). Since the magnitude is constant (it doesn't depend on t), the integral from 0 to 1 is just ( sqrt{38} times (1 - 0) = sqrt{38} ). So the length of Segment 1 is ( sqrt{38} ).Moving on to Segment 2: ( mathbf{r}_2(u) = (2 + 4u, 3 + 2u, 5 - u) ) for ( u ) in [0, 1].Again, I'll find the derivative with respect to u. The derivative, ( mathbf{r}_2'(u) ), is (4, 2, -1). Calculating the magnitude: ( sqrt{4^2 + 2^2 + (-1)^2} = sqrt{16 + 4 + 1} = sqrt{21} ).Since this magnitude is also constant, the length of Segment 2 is ( sqrt{21} times (1 - 0) = sqrt{21} ).Now, Segment 3: ( mathbf{r}_3(v) = (6 + v, 5 + 3v, 4 + 2v) ) for ( v ) in [0, 1].Derivative with respect to v: ( mathbf{r}_3'(v) = (1, 3, 2) ).Magnitude: ( sqrt{1^2 + 3^2 + 2^2} = sqrt{1 + 9 + 4} = sqrt{14} ).So, the length of Segment 3 is ( sqrt{14} times (1 - 0) = sqrt{14} ).To find the total length, I'll add up the lengths of all three segments:Total length = ( sqrt{38} + sqrt{21} + sqrt{14} ).Hmm, let me compute these square roots numerically to see if it makes sense. ( sqrt{38} ) is approximately 6.164, ( sqrt{21} ) is about 4.583, and ( sqrt{14} ) is roughly 3.741. Adding them up: 6.164 + 4.583 = 10.747; 10.747 + 3.741 = 14.488. So the total length is approximately 14.488 units. But I should keep it in exact form unless specified otherwise. So, the total length is ( sqrt{38} + sqrt{21} + sqrt{14} ).Wait, before I proceed, let me double-check my calculations. For each segment, I took the derivative, found the magnitude, and since each magnitude was constant, I just multiplied by the interval length, which is 1 in each case. That seems correct.Now, moving on to the second part: finding the total variation in altitude throughout the entire flight path. The altitude is given by the z-coordinate, so I need to consider the z-component of each segment and find the total variation.Total variation is the sum of the absolute differences in altitude over each segment. So, for each segment, I'll compute the change in z from the start to the end and take the absolute value, then add them all together.Starting with Segment 1: ( z(t) = 5t ). At t=0, z=0; at t=1, z=5. So the change is 5 - 0 = 5. Absolute value is 5.Segment 2: ( z(u) = 5 - u ). At u=0, z=5; at u=1, z=4. So the change is 4 - 5 = -1. Absolute value is 1.Segment 3: ( z(v) = 4 + 2v ). At v=0, z=4; at v=1, z=6. So the change is 6 - 4 = 2. Absolute value is 2.Adding these up: 5 + 1 + 2 = 8.Wait, hold on. Is that all? Or do I need to consider the variation within each segment, not just the endpoints? Hmm, total variation is the sum of the absolute changes in altitude over each segment. Since each segment is a straight line, the altitude changes monotonically, so the total variation is just the sum of the absolute differences at the endpoints.But let me think again. For a function, total variation is the supremum of the sum of absolute differences over all possible partitions. But since each segment is a straight line, the altitude is a linear function, so the maximum variation is just the difference between the start and end points. So yes, adding the absolute differences at the endpoints for each segment should give the total variation.Therefore, the total variation in altitude is 5 + 1 + 2 = 8 units.Wait, but let me verify for each segment:Segment 1: z goes from 0 to 5, so variation is 5.Segment 2: z goes from 5 to 4, so variation is 1.Segment 3: z goes from 4 to 6, so variation is 2.Total: 5 + 1 + 2 = 8. That seems correct.Alternatively, if I were to compute the total variation over the entire path, it's the same as the sum of the variations on each segment because each segment is traversed in one direction without any backtracking in altitude. So, yes, 8 is correct.So, summarizing:1. Total flight path length: ( sqrt{38} + sqrt{21} + sqrt{14} ).2. Total variation in altitude: 8.I think that's it. Let me just make sure I didn't make any calculation errors.For Segment 1:- Derivative: (2,3,5), magnitude ( sqrt{4 + 9 + 25} = sqrt{38} ). Correct.Segment 2:- Derivative: (4,2,-1), magnitude ( sqrt{16 + 4 + 1} = sqrt{21} ). Correct.Segment 3:- Derivative: (1,3,2), magnitude ( sqrt{1 + 9 + 4} = sqrt{14} ). Correct.Altitude changes:- Segment 1: 0 to 5, change 5.- Segment 2: 5 to 4, change 1.- Segment 3: 4 to 6, change 2.Total variation: 5 + 1 + 2 = 8. Correct.So, I think my answers are correct.Final Answer1. The total length of the flight path is boxed{sqrt{38} + sqrt{21} + sqrt{14}}.2. The total variation in altitude is boxed{8}.</think>