answer：First, I need to determine the total number of hours the couple spends each week on the three activities recommended by Dr. Taylor. They spend 5 hours on communication exercises, 3 hours on financial planning, and 2 hours on relaxation activities. Adding these together gives a weekly total of 10 hours.Next, since the couple follows this routine for 4 weeks, I will multiply the weekly total by 4 to find the overall number of hours spent on these activities.Finally, calculating 10 hours multiplied by 4 weeks results in a total of 40 hours.

answer：Okay, so I've got this problem about a modified Fibonacci sequence assigned to the faces of a regular dodecahedron. The professor is honoring Giustino Ferri, who was a mathematician, so this seems like a nice blend of geometry and number theory. Let me try to break it down step by step.First, the problem mentions a regular dodecahedron, which I remember is one of the Platonic solids. It has 12 regular pentagonal faces. So, each face is going to be assigned a unique integer from this special sequence. The sequence is inspired by Fibonacci but with a twist: every third number is multiplied by 11, which is Ferri's favorite prime number.Alright, so let's tackle Problem 1 first: finding the sum of the numbers assigned to the 12 faces. That means I need to generate the first 12 numbers of this modified Fibonacci sequence and then add them up.The standard Fibonacci sequence starts with 1 and 1, and each subsequent number is the sum of the two preceding ones. So, the sequence goes 1, 1, 2, 3, 5, 8, 13, 21, and so on. But in this case, every third number is multiplied by 11. Hmm, so starting from the third term, every third term gets multiplied by 11.Let me write out the sequence step by step. I'll number the terms from 1 to 12.Term 1: 1 (given)Term 2: 1 (given)Term 3: Term1 + Term2 = 1 + 1 = 2. Since it's the third term, multiply by 11: 2 * 11 = 22Term 4: Term2 + Term3 = 1 + 22 = 23Term 5: Term3 + Term4 = 22 + 23 = 45Term 6: Term4 + Term5 = 23 + 45 = 68. Since it's the sixth term, multiply by 11: 68 * 11 = 748Term 7: Term5 + Term6 = 45 + 748 = 793Term 8: Term6 + Term7 = 748 + 793 = 1541Term 9: Term7 + Term8 = 793 + 1541 = 2334. Since it's the ninth term, multiply by 11: 2334 * 11 = 25674Term 10: Term8 + Term9 = 1541 + 25674 = 27215Term 11: Term9 + Term10 = 25674 + 27215 = 52889Term 12: Term10 + Term11 = 27215 + 52889 = 80104. Since it's the twelfth term, multiply by 11: 80104 * 11 = 881144Wait, hold on. Let me double-check the terms because the multiplication by 11 is every third term starting from term 3. So term 3, 6, 9, 12, etc., are multiplied by 11. So, in my calculation above, term 3, 6, 9, and 12 are multiplied by 11. Let me verify each term:Term1: 1Term2: 1Term3: (1+1)=2, multiplied by 11: 22Term4: Term2 + Term3 = 1 + 22 = 23Term5: Term3 + Term4 = 22 + 23 = 45Term6: Term4 + Term5 = 23 + 45 = 68, multiplied by 11: 748Term7: Term5 + Term6 = 45 + 748 = 793Term8: Term6 + Term7 = 748 + 793 = 1541Term9: Term7 + Term8 = 793 + 1541 = 2334, multiplied by 11: 25674Term10: Term8 + Term9 = 1541 + 25674 = 27215Term11: Term9 + Term10 = 25674 + 27215 = 52889Term12: Term10 + Term11 = 27215 + 52889 = 80104, multiplied by 11: 881144Okay, that seems correct. So now, I need to list all 12 terms:1, 1, 22, 23, 45, 748, 793, 1541, 25674, 27215, 52889, 881144Now, I need to sum these up. Let me add them step by step:Start with 1 + 1 = 22 + 22 = 2424 + 23 = 4747 + 45 = 9292 + 748 = 840840 + 793 = 16331633 + 1541 = 31743174 + 25674 = 2884828848 + 27215 = 5606356063 + 52889 = 108,952108,952 + 881,144 = 990,096Wait, let me verify that addition step by step to make sure I didn't make a mistake:1 + 1 = 22 + 22 = 2424 + 23 = 4747 + 45 = 9292 + 748 = 840840 + 793 = 16331633 + 1541 = 31743174 + 25674 = 2884828848 + 27215 = 5606356063 + 52889 = 108,952108,952 + 881,144 = 990,096Hmm, that seems quite large. Let me check the individual terms again to make sure I didn't miscalculate any of them.Term1: 1Term2: 1Term3: 2*11=22Term4: 1+22=23Term5:22+23=45Term6:23+45=68*11=748Term7:45+748=793Term8:748+793=1541Term9:793+1541=2334*11=25674Term10:1541+25674=27215Term11:25674+27215=52889Term12:27215+52889=80104*11=881144Yes, all terms seem correct. So adding them up:1 + 1 = 22 + 22 = 2424 + 23 = 4747 + 45 = 9292 + 748 = 840840 + 793 = 16331633 + 1541 = 31743174 + 25674 = 2884828848 + 27215 = 5606356063 + 52889 = 108,952108,952 + 881,144 = 990,096So, the sum S is 990,096.Wait, that seems enormous. Let me cross-verify the addition:Let me list all terms:1, 1, 22, 23, 45, 748, 793, 1541, 25674, 27215, 52889, 881144Let me add them in pairs to make it easier:1 + 881144 = 8811451 + 52889 = 5289022 + 27215 = 2723723 + 25674 = 2569745 + 1541 = 1586748 + 793 = 1541So now, we have the pairs:881145, 52890, 27237, 25697, 1586, 1541Now, let's add these:Start with 881,145 + 52,890 = 934,035934,035 + 27,237 = 961,272961,272 + 25,697 = 986,969986,969 + 1,586 = 988,555988,555 + 1,541 = 990,096Okay, same result. So, S = 990,096.Alright, so that's Problem 1 done. Now, moving on to Problem 2: Determine if there exist integer solutions for x and y in the equation x² + y² = S, which is 990,096. If so, find one such pair (x, y).Hmm, so we need to check if 990,096 can be expressed as the sum of two squares. I remember that a number can be expressed as the sum of two squares if and only if in its prime factorization, every prime congruent to 3 modulo 4 appears with an even exponent.So, first, let's factorize 990,096.But 990,096 is a large number. Let me see if I can factor it step by step.First, note that 990,096 is even, so divide by 2:990,096 ÷ 2 = 495,048495,048 ÷ 2 = 247,524247,524 ÷ 2 = 123,762123,762 ÷ 2 = 61,88161,881 is odd. Let's check divisibility by 3: 6 + 1 + 8 + 8 + 1 = 24, which is divisible by 3.61,881 ÷ 3 = 20,627Now, 20,627. Let's check divisibility by small primes:20,627 ÷ 7 = 2,946.714... Not integer.20,627 ÷ 11 = 1,875.181... Not integer.20,627 ÷ 13 = 1,586.692... Not integer.20,627 ÷ 17 = 1,213.352... Not integer.20,627 ÷ 19 = 1,085.631... Not integer.20,627 ÷ 23 = 896.826... Not integer.20,627 ÷ 29 = 711.275... Not integer.20,627 ÷ 31 = 665.387... Not integer.20,627 ÷ 37 = 557.486... Not integer.20,627 ÷ 43 = 479.697... Not integer.20,627 ÷ 47 = 438.872... Not integer.20,627 ÷ 53 = 390.320... Not integer.Hmm, maybe 20,627 is a prime? Let me check.Wait, 20,627. Let me try dividing by 7: 7*2946=20,622, remainder 5. Not divisible by 7.Divide by 11: 11*1875=20,625, remainder 2. Not divisible by 11.Divide by 13: 13*1586=20,618, remainder 9. Not divisible by 13.Divide by 17: 17*1213=20,621, remainder 6. Not divisible by 17.Divide by 19: 19*1085=20,615, remainder 12. Not divisible by 19.Divide by 23: 23*896=20,608, remainder 19. Not divisible by 23.Divide by 29: 29*711=20,619, remainder 8. Not divisible by 29.Divide by 31: 31*665=20,615, remainder 12. Not divisible by 31.Divide by 37: 37*557=20,609, remainder 18. Not divisible by 37.Divide by 43: 43*479=20,637, which is higher. So, remainder negative. Not divisible.Divide by 47: 47*438=20,686, which is higher. Not divisible.So, it seems 20,627 is a prime number. Let me confirm with a primality test.Alternatively, maybe I made a mistake earlier in factorization.Wait, let's go back:We had 990,096.Divided by 2 four times: 990,096 / 16 = 61,881.61,881 divided by 3 is 20,627.So, 990,096 = 2^4 * 3 * 20,627.Now, if 20,627 is prime, then the prime factors are 2, 3, and 20,627.Now, to check if 20,627 is prime, let's see if it's divisible by primes up to sqrt(20,627). The square root of 20,627 is approximately 143.6.So, we need to check primes up to 143.We checked up to 47 earlier. Let's continue:53: 53*390=20,670, which is higher. 53*389=20,617, remainder 10. Not divisible.59: 59*349=20,691, which is higher. 59*349=20,691, so 59*349-20,627=64. Not divisible.61: 61*338=20,618, remainder 9. Not divisible.67: 67*307=20,569, remainder 58. Not divisible.71: 71*290=20,590, remainder 37. Not divisible.73: 73*282=20,586, remainder 41. Not divisible.79: 79*260=20,540, remainder 87. 87 is divisible by 79? No, 79*1=79, remainder 8. Not divisible.83: 83*248=20,584, remainder 43. Not divisible.89: 89*231=20,559, remainder 68. Not divisible.97: 97*212=20,564, remainder 63. Not divisible.101: 101*204=20,604, remainder 23. Not divisible.103: 103*200=20,600, remainder 27. Not divisible.107: 107*192=20,544, remainder 83. Not divisible.109: 109*189=20,501, remainder 126. Not divisible.113: 113*182=20,516, remainder 111. Not divisible.127: 127*162=20,574, remainder 53. Not divisible.131: 131*157=20,567, remainder 60. Not divisible.137: 137*150=20,550, remainder 77. Not divisible.139: 139*148=20,572, remainder 55. Not divisible.149 is beyond sqrt(20,627), which is ~143.6, so we can stop here.Since none of the primes up to 143 divide 20,627, it must be a prime number.Therefore, the prime factorization of 990,096 is 2^4 * 3 * 20,627.Now, to check if 990,096 can be expressed as the sum of two squares, we need to look at its prime factors.The rule is that a number can be expressed as the sum of two squares if and only if in its prime factorization, every prime congruent to 3 mod 4 appears with an even exponent.Looking at the prime factors:2: exponent 4 (which is even)3: exponent 1 (3 mod 4 is 3, and exponent is 1, which is odd)20,627: Let's check if it's congruent to 3 mod 4.20,627 ÷ 4 = 5,156.75, so remainder 3. So, 20,627 ≡ 3 mod 4.Exponent of 20,627 is 1, which is odd.So, both primes 3 and 20,627 are congruent to 3 mod 4 and have odd exponents. Therefore, according to the theorem, 990,096 cannot be expressed as the sum of two squares.Wait, but hold on. The theorem says that if any prime congruent to 3 mod 4 has an odd exponent, then the number cannot be expressed as the sum of two squares. Since we have two such primes with odd exponents, the conclusion is that it's not expressible.But wait, let me double-check. Maybe I made a mistake in the factorization.Wait, 20,627 is a prime, and it's 3 mod 4. So, in the factorization, we have 3^1 * 20,627^1, both primes congruent to 3 mod 4 with odd exponents. Therefore, the number cannot be expressed as the sum of two squares.But wait, let me think again. The theorem says that in the prime factorization, every prime congruent to 3 mod 4 must have an even exponent. Here, both 3 and 20,627 are 3 mod 4 and have exponent 1, which is odd. Therefore, the number cannot be expressed as the sum of two squares.Therefore, the answer to Problem 2 is that there are no integer solutions (x, y) such that x² + y² = 990,096.Wait, but let me check if I did everything correctly. Maybe I missed something.Alternatively, perhaps I made a mistake in the factorization. Let me try another approach.Wait, 990,096 is equal to 12 * 82,508. Wait, no, 12*82,508=990,096. But 82,508 is 4*20,627. So, 990,096=12*82,508=12*4*20,627=48*20,627. Hmm, but that's consistent with earlier factorization.Alternatively, maybe I can try to see if 990,096 is a perfect square. Let's see, sqrt(990,096) ≈ 995.04. So, 995²=990,025, and 996²=992,016. So, 990,096 is between these two squares. Therefore, it's not a perfect square, but that doesn't necessarily mean it can't be expressed as the sum of two squares.Wait, but according to the theorem, since it has primes congruent to 3 mod 4 with odd exponents, it cannot be expressed as the sum of two squares. So, I think my conclusion is correct.Therefore, Problem 2 has no solution.Wait, but let me try another approach. Maybe I can use the fact that if a number is divisible by 4, then it can be expressed as the sum of two squares if and only if the quotient is expressible as such. Wait, 990,096 is divisible by 16 (since we divided by 2 four times). So, 990,096 = 16 * 61,881. Now, 61,881 is 3 * 20,627. So, 61,881 is 3 mod 4, as 61,881 ÷ 4 = 15,470.25, so remainder 1. Wait, no, 61,881 ÷ 4 is 15,470 with a remainder of 1, so 61,881 ≡ 1 mod 4.Wait, but 61,881 is 3 * 20,627, both primes congruent to 3 mod 4. So, 61,881 is 3 * 20,627, which is (3 mod 4)*(3 mod 4)=9 mod 4=1 mod 4. So, 61,881 ≡1 mod4.But 61,881 is 3 * 20,627, both primes congruent to 3 mod4, each with exponent 1, which is odd. Therefore, 61,881 cannot be expressed as the sum of two squares. Therefore, 990,096=16*61,881, and since 61,881 cannot be expressed as the sum of two squares, neither can 990,096.Wait, but actually, the rule is that if a number is divisible by 4, then it can be expressed as the sum of two squares if and only if the quotient after dividing by 4 can be expressed as the sum of two squares. But in this case, 61,881 cannot be expressed as the sum of two squares, so neither can 990,096.Therefore, the conclusion is correct: there are no integer solutions (x, y) such that x² + y² = 990,096.Wait, but let me try to see if there's a way to express it as a sum of two squares despite the theorem. Maybe I made a mistake in the factorization.Alternatively, perhaps I can use the fact that 990,096 is divisible by 16, so let me check if 61,881 can be expressed as the sum of two squares. If not, then 990,096 can't either.But 61,881 is 3 * 20,627, both primes congruent to 3 mod4, each with exponent 1. Therefore, 61,881 cannot be expressed as the sum of two squares, so 990,096 can't either.Therefore, the answer to Problem 2 is that there are no integer solutions.Wait, but let me double-check the factorization once more to be sure.990,096 ÷ 2 = 495,048495,048 ÷ 2 = 247,524247,524 ÷ 2 = 123,762123,762 ÷ 2 = 61,88161,881 ÷ 3 = 20,62720,627 is prime, as we checked earlier.So, 990,096 = 2^4 * 3 * 20,627.Yes, that's correct.Therefore, since both 3 and 20,627 are primes congruent to 3 mod4 with odd exponents, the number cannot be expressed as the sum of two squares.So, Problem 2 has no solution.Wait, but just to be thorough, let me try to see if there's any possible way to express 990,096 as the sum of two squares. Maybe using some computational method.But given the size of the number, it's impractical to check manually. However, based on the theorem, it's impossible. So, I think the conclusion is solid.Therefore, the answers are:Problem 1: S = 990,096Problem 2: No integer solutions exist for x and y.But wait, let me think again. Maybe I made a mistake in the initial sequence generation. Let me recheck the terms:Term1:1Term2:1Term3:1+1=2*11=22Term4:1+22=23Term5:22+23=45Term6:23+45=68*11=748Term7:45+748=793Term8:748+793=1541Term9:793+1541=2334*11=25674Term10:1541+25674=27215Term11:25674+27215=52889Term12:27215+52889=80104*11=881,144Yes, that's correct. So the sum is indeed 990,096.Therefore, the conclusion for Problem 2 is correct.**Final Answer**Problem 1: boxed{990096}Problem 2: No integer solutions exist.

answer：Okay, so I have this problem where I need to find the time ( t ) when the number of readers for two book series, Series H and Series S, will be equal. The functions given are ( R_H(t) = 50 + 10t + 0.5t^2 ) and ( R_S(t) = 20e^{0.1t} ). First, for part (a), I need to set these two functions equal to each other and solve for ( t ). That means I have to solve the equation:[ 50 + 10t + 0.5t^2 = 20e^{0.1t} ]Hmm, this looks like a transcendental equation because it involves both polynomial terms and an exponential term. I remember that transcendental equations can't usually be solved algebraically, so I might need to use numerical methods or graphing to find the solution.Let me write down the equation again:[ 0.5t^2 + 10t + 50 = 20e^{0.1t} ]Maybe I can rearrange it to bring all terms to one side:[ 0.5t^2 + 10t + 50 - 20e^{0.1t} = 0 ]Let me define a function ( f(t) = 0.5t^2 + 10t + 50 - 20e^{0.1t} ). So, I need to find the root of ( f(t) = 0 ).To find the root, I can use methods like the Newton-Raphson method or the bisection method. But since I don't have a calculator here, maybe I can estimate it by plugging in some values of ( t ) and see where ( f(t) ) crosses zero.Let me try some values:When ( t = 0 ):( f(0) = 0 + 0 + 50 - 20e^{0} = 50 - 20 = 30 ). So, positive.When ( t = 10 ):( f(10) = 0.5*(100) + 100 + 50 - 20e^{1} )= 50 + 100 + 50 - 20*2.718= 200 - 54.36= 145.64. Still positive.Wait, that's still positive. Maybe I need a larger ( t ).Wait, let me check ( t = 20 ):( f(20) = 0.5*(400) + 200 + 50 - 20e^{2} )= 200 + 200 + 50 - 20*7.389= 450 - 147.78= 302.22. Still positive.Hmm, maybe even larger. Let's try ( t = 30 ):( f(30) = 0.5*(900) + 300 + 50 - 20e^{3} )= 450 + 300 + 50 - 20*20.085= 800 - 401.7= 398.3. Still positive.Wait, this is confusing. Maybe I made a mistake in my calculations.Wait, let's check ( t = 10 ) again:( f(10) = 0.5*(10)^2 + 10*(10) + 50 - 20e^{0.1*10} )= 0.5*100 + 100 + 50 - 20e^{1}= 50 + 100 + 50 - 20*2.718= 200 - 54.36= 145.64. Correct.Wait, but the exponential function ( e^{0.1t} ) grows faster than the quadratic function eventually, right? So maybe after a certain point, the exponential will overtake the quadratic. But according to my calculations, even at ( t = 30 ), the quadratic is still higher.Wait, let me check ( t = 50 ):( f(50) = 0.5*(2500) + 500 + 50 - 20e^{5} )= 1250 + 500 + 50 - 20*148.413= 1800 - 2968.26= -1168.26. Now it's negative.So, somewhere between ( t = 30 ) and ( t = 50 ), the function crosses zero. Let me narrow it down.Let me try ( t = 40 ):( f(40) = 0.5*(1600) + 400 + 50 - 20e^{4} )= 800 + 400 + 50 - 20*54.598= 1250 - 1091.96= 158.04. Still positive.So between 40 and 50.At ( t = 45 ):( f(45) = 0.5*(2025) + 450 + 50 - 20e^{4.5} )= 1012.5 + 450 + 50 - 20*90.017= 1512.5 - 1800.34= -287.84. Negative.So between 40 and 45.At ( t = 43 ):( f(43) = 0.5*(1849) + 430 + 50 - 20e^{4.3} )= 924.5 + 430 + 50 - 20*73.726= 1404.5 - 1474.52= -70.02. Negative.At ( t = 42 ):( f(42) = 0.5*(1764) + 420 + 50 - 20e^{4.2} )= 882 + 420 + 50 - 20*66.765= 1352 - 1335.3= 16.7. Positive.So between 42 and 43.At ( t = 42.5 ):( f(42.5) = 0.5*(42.5)^2 + 10*42.5 + 50 - 20e^{0.1*42.5} )First, calculate each term:( 0.5*(42.5)^2 = 0.5*1806.25 = 903.125 )( 10*42.5 = 425 )So, 903.125 + 425 + 50 = 1378.125Now, ( e^{0.1*42.5} = e^{4.25} approx 69.07 )So, 20*69.07 = 1381.4Thus, ( f(42.5) = 1378.125 - 1381.4 ≈ -3.275 ). Negative.So between 42 and 42.5.At ( t = 42.25 ):Calculate ( f(42.25) ):( 0.5*(42.25)^2 = 0.5*(1785.0625) = 892.53125 )( 10*42.25 = 422.5 )So, 892.53125 + 422.5 + 50 = 1365.03125( e^{0.1*42.25} = e^{4.225} ≈ e^{4.2} * e^{0.025} ≈ 66.765 * 1.0253 ≈ 68.46 )So, 20*68.46 ≈ 1369.2Thus, ( f(42.25) ≈ 1365.03 - 1369.2 ≈ -4.17 ). Still negative.Wait, but at t=42, f(t)=16.7, which is positive, and at t=42.25, f(t)=-4.17. So the root is between 42 and 42.25.Let me try t=42.1:( f(42.1) = 0.5*(42.1)^2 + 10*42.1 + 50 - 20e^{0.1*42.1} )Calculate each term:( 42.1^2 = 1772.41 )So, 0.5*1772.41 = 886.20510*42.1 = 421So, 886.205 + 421 + 50 = 1357.205Now, ( e^{4.21} ≈ e^{4.2} * e^{0.01} ≈ 66.765 * 1.01005 ≈ 67.44 )So, 20*67.44 ≈ 1348.8Thus, ( f(42.1) ≈ 1357.205 - 1348.8 ≈ 8.405 ). Positive.So between 42.1 and 42.25.At t=42.2:( f(42.2) = 0.5*(42.2)^2 + 10*42.2 + 50 - 20e^{4.22} )Calculate:42.2^2 = 1780.840.5*1780.84 = 890.4210*42.2 = 422So, 890.42 + 422 + 50 = 1362.42( e^{4.22} ≈ e^{4.2} * e^{0.02} ≈ 66.765 * 1.0202 ≈ 68.12 )20*68.12 ≈ 1362.4Thus, ( f(42.2) ≈ 1362.42 - 1362.4 ≈ 0.02 ). Almost zero.Wow, that's very close. So at t=42.2, f(t)≈0.02, which is almost zero. Let me check t=42.2:Wait, let me recalculate ( e^{4.22} ) more accurately.Using a calculator, e^4.22 ≈ e^4 * e^0.22 ≈ 54.598 * 1.2461 ≈ 54.598*1.2461 ≈ let's compute:54.598 * 1 = 54.59854.598 * 0.2 = 10.919654.598 * 0.04 = 2.1839254.598 * 0.0061 ≈ 0.333Adding up: 54.598 + 10.9196 = 65.5176 + 2.18392 = 67.7015 + 0.333 ≈ 68.0345So, e^4.22 ≈ 68.0345Thus, 20*e^4.22 ≈ 20*68.0345 ≈ 1360.69So, f(42.2) = 1362.42 - 1360.69 ≈ 1.73. Wait, that contradicts my earlier calculation. Hmm, maybe I made a mistake in the previous step.Wait, let me recalculate f(42.2):0.5*(42.2)^2 = 0.5*(1780.84) = 890.4210*42.2 = 422So, 890.42 + 422 + 50 = 1362.4220*e^{4.22} ≈ 20*68.0345 ≈ 1360.69Thus, f(42.2) = 1362.42 - 1360.69 ≈ 1.73. Positive.Wait, so at t=42.2, f(t)=1.73.Wait, earlier at t=42.25, f(t)=-4.17. So between 42.2 and 42.25, f(t) goes from positive to negative.Wait, but at t=42.2, f(t)=1.73, and at t=42.25, f(t)=-4.17.So, the root is between 42.2 and 42.25.Let me try t=42.22:Calculate f(42.22):0.5*(42.22)^2 = 0.5*(1782.5284) ≈ 891.264210*42.22 = 422.2So, 891.2642 + 422.2 + 50 ≈ 1363.4642Now, e^{0.1*42.22} = e^{4.222} ≈ e^{4.22} * e^{0.002} ≈ 68.0345 * 1.002001 ≈ 68.0345 + 0.136 ≈ 68.1705Thus, 20*e^{4.222} ≈ 20*68.1705 ≈ 1363.41So, f(42.22) ≈ 1363.4642 - 1363.41 ≈ 0.0542. Almost zero.That's very close. So, f(42.22)≈0.0542.Now, let's try t=42.23:e^{4.223} ≈ e^{4.22} * e^{0.003} ≈ 68.0345 * 1.0030045 ≈ 68.0345 + 0.204 ≈ 68.238520*e^{4.223} ≈ 20*68.2385 ≈ 1364.77f(42.23) = 0.5*(42.23)^2 + 10*42.23 + 50 - 20e^{4.223}Calculate 0.5*(42.23)^2:42.23^2 = (42 + 0.23)^2 = 42^2 + 2*42*0.23 + 0.23^2 = 1764 + 19.32 + 0.0529 ≈ 1783.37290.5*1783.3729 ≈ 891.6864510*42.23 = 422.3So, 891.68645 + 422.3 + 50 ≈ 1363.9864520*e^{4.223} ≈ 1364.77Thus, f(42.23) ≈ 1363.98645 - 1364.77 ≈ -0.78355. Negative.So, between t=42.22 and t=42.23, f(t) crosses zero.At t=42.22, f(t)=0.0542At t=42.23, f(t)=-0.78355So, using linear approximation:The change in t is 0.01, and the change in f(t) is -0.78355 - 0.0542 ≈ -0.83775We need to find the t where f(t)=0, starting from t=42.22.The fraction needed is 0.0542 / 0.83775 ≈ 0.0647So, t ≈ 42.22 + 0.0647*0.01 ≈ 42.22 + 0.000647 ≈ 42.2206So, approximately t≈42.22 months.But let me check at t=42.22:f(t)=0.0542At t=42.2206:The change in t is 0.0006, so the change in f(t) is approximately derivative at t=42.22 times 0.0006.But maybe it's too small. Alternatively, since f(t) at 42.22 is 0.0542, and at 42.2206, it's approximately 0.0542 - (0.83775/0.01)*0.0006 ≈ 0.0542 - 0.83775*0.06 ≈ 0.0542 - 0.0503 ≈ 0.0039. Still positive.Wait, maybe I should use linear approximation between t=42.22 and t=42.23.At t1=42.22, f1=0.0542At t2=42.23, f2=-0.78355The root is at t = t1 - f1*(t2 - t1)/(f2 - f1)So, t = 42.22 - 0.0542*(0.01)/(-0.78355 - 0.0542)= 42.22 - 0.0542*0.01/(-0.83775)= 42.22 + (0.0542*0.01)/0.83775= 42.22 + 0.000542/0.83775≈ 42.22 + 0.000647≈ 42.220647So, approximately t≈42.2206 months.To check, let's compute f(42.2206):0.5*(42.2206)^2 + 10*42.2206 + 50 - 20e^{0.1*42.2206}First, 42.2206^2 ≈ (42.22)^2 + 2*42.22*0.0006 + (0.0006)^2 ≈ 1783.3729 + 0.0507 + 0.00000036 ≈ 1783.42360.5*1783.4236 ≈ 891.711810*42.2206 ≈ 422.206So, 891.7118 + 422.206 + 50 ≈ 1363.9178Now, e^{4.22206} ≈ e^{4.22} * e^{0.00206} ≈ 68.0345 * 1.00206 ≈ 68.0345 + 0.1398 ≈ 68.174320*e^{4.22206} ≈ 20*68.1743 ≈ 1363.486Thus, f(42.2206) ≈ 1363.9178 - 1363.486 ≈ 0.4318. Wait, that's not zero. Hmm, maybe my approximation is off.Wait, perhaps I made a mistake in calculating e^{4.22206}. Let me compute e^{4.22206} more accurately.Using a calculator, e^4.22206 ≈ e^4.22 * e^0.00206 ≈ 68.0345 * 1.00206 ≈ 68.0345 + 68.0345*0.00206 ≈ 68.0345 + 0.140 ≈ 68.1745So, 20*e^{4.22206} ≈ 1363.49Thus, f(42.2206) ≈ 1363.9178 - 1363.49 ≈ 0.4278. Still positive.Wait, so my earlier approximation was not accurate enough. Maybe I need to do another iteration.Let me use the Newton-Raphson method.Given f(t) = 0.5t² + 10t + 50 - 20e^{0.1t}f'(t) = t + 10 - 2e^{0.1t}At t=42.22, f(t)=0.0542f'(42.22)=42.22 +10 -2e^{4.222}≈52.22 -2*68.1745≈52.22 -136.349≈-84.129So, Newton-Raphson update:t1 = t0 - f(t0)/f'(t0) = 42.22 - 0.0542/(-84.129) ≈42.22 + 0.000644≈42.220644So, t≈42.220644Now, compute f(42.220644):0.5*(42.220644)^2 +10*42.220644 +50 -20e^{0.1*42.220644}Calculate 42.220644^2:≈ (42.22)^2 + 2*42.22*0.000644 + (0.000644)^2 ≈1783.3729 + 0.0542 + 0.000000415≈1783.42710.5*1783.4271≈891.7135510*42.220644≈422.20644So, 891.71355 + 422.20644 +50≈1363.92Now, e^{4.2220644}≈e^{4.22} * e^{0.0020644}≈68.0345 *1.002067≈68.0345 +0.1398≈68.174320*e^{4.2220644}≈1363.486Thus, f(t)=1363.92 -1363.486≈0.434Wait, that's still positive. Hmm, maybe I need to do another iteration.Compute f'(42.220644)=42.220644 +10 -2e^{4.2220644}≈52.220644 -2*68.1743≈52.220644 -136.3486≈-84.128So, t1=42.220644 -0.434/(-84.128)≈42.220644 +0.00516≈42.2258Wait, but at t=42.2258, f(t)=?Wait, but earlier at t=42.22, f(t)=0.0542, and at t=42.2206, f(t)=0.434. That doesn't make sense because increasing t should decrease f(t) since the exponential is growing faster.Wait, perhaps I made a mistake in the calculation.Wait, let me recalculate f(42.220644):0.5*(42.220644)^2 =0.5*(42.220644*42.220644)Let me compute 42.220644*42.220644:42*42=176442*0.220644=9.2670.220644*42=9.2670.220644*0.220644≈0.04868So, total≈1764 +9.267 +9.267 +0.04868≈1782.58268Wait, that's not matching my earlier calculation. Maybe I should use a calculator approach.Alternatively, perhaps I should use a better method.Alternatively, maybe I should accept that t≈42.22 months is close enough, given that the function is crossing zero there.But wait, earlier at t=42.22, f(t)=0.0542, and at t=42.2206, f(t)=0.434. That seems inconsistent because increasing t should decrease f(t) as the exponential term grows faster.Wait, perhaps I made a mistake in calculating f(t) at t=42.2206.Wait, let me compute e^{4.2220644} more accurately.Using a calculator, e^4.2220644 ≈ e^4.222 ≈ let's compute:We know that e^4=54.59815e^0.222≈1.24889So, e^4.222≈54.59815*1.24889≈54.59815*1.2=65.51778, 54.59815*0.04889≈2.666, so total≈65.51778+2.666≈68.1838Thus, e^4.222≈68.1838So, 20*e^{4.222}≈1363.676Now, f(t)=0.5t² +10t +50 -20e^{0.1t}At t=42.220644:0.5*(42.220644)^2≈0.5*(1783.427)≈891.713510*42.220644≈422.20644So, 891.7135 +422.20644 +50≈1363.92Thus, f(t)=1363.92 -1363.676≈0.244So, f(t)=0.244 at t=42.220644Now, f'(t)=t +10 -2e^{0.1t}=42.220644 +10 -2*68.1838≈52.220644 -136.3676≈-84.147So, Newton-Raphson update:t1 = t0 - f(t0)/f'(t0)=42.220644 -0.244/(-84.147)=42.220644 +0.002898≈42.223542Now, compute f(42.223542):0.5*(42.223542)^2 +10*42.223542 +50 -20e^{4.2223542}Calculate 42.223542^2:≈(42.22)^2 + 2*42.22*0.003542 + (0.003542)^2≈1783.3729 +0.300 +0.0000125≈1783.67390.5*1783.6739≈891.8369510*42.223542≈422.23542So, 891.83695 +422.23542 +50≈1364.07237Now, e^{4.2223542}≈e^{4.222}≈68.1838 (as before, since 4.2223542 is very close to 4.222)Thus, 20*e^{4.2223542}≈1363.676So, f(t)=1364.07237 -1363.676≈0.39637Wait, that's still positive. Hmm, this suggests that my approximation is not converging as expected. Maybe I need to use a better method or accept that t≈42.22 months is the approximate solution.Alternatively, perhaps I should use a calculator or software to find a more accurate value, but since I'm doing this manually, I'll settle for t≈42.22 months.For part (b), once I have t≈42.22 months, I can plug it back into either R_H(t) or R_S(t) to find the reader count.Let's use R_H(t):R_H(42.22)=50 +10*42.22 +0.5*(42.22)^2=50 +422.2 +0.5*(1783.3729)=50 +422.2 +891.68645=50 +422.2=472.2 +891.68645≈1363.88645 thousand readers.Alternatively, using R_S(t):R_S(42.22)=20e^{0.1*42.22}=20e^{4.222}≈20*68.1838≈1363.676 thousand readers.So, approximately 1363.7 thousand readers.But since the functions are equal at this point, the exact value would be the same for both.So, rounding to a reasonable decimal place, perhaps t≈42.22 months and R(t)≈1363.7 thousand readers.But let me check if I can get a more accurate t.Alternatively, perhaps I can use the fact that at t=42.22, f(t)=0.0542, and at t=42.2206, f(t)=0.244, which suggests that my earlier approximation was off.Alternatively, perhaps I should use the secant method between t=42.22 and t=42.23.At t1=42.22, f1=0.0542At t2=42.23, f2=-0.78355The secant method formula is:t3 = t2 - f2*(t2 - t1)/(f2 - f1)=42.23 - (-0.78355)*(0.01)/(-0.78355 -0.0542)=42.23 + (0.78355*0.01)/(-0.83775)=42.23 - (0.0078355)/0.83775≈42.23 -0.00935≈42.22065So, t≈42.22065Thus, t≈42.2207 months.So, the answer for part (a) is approximately t≈42.22 months, and for part (b), R(t)≈1363.7 thousand readers.But to be more precise, let me compute R(t) at t=42.2207:R_H(t)=50 +10*42.2207 +0.5*(42.2207)^2=50 +422.207 +0.5*(1783.427)=50 +422.207 +891.7135≈1363.9205R_S(t)=20e^{0.1*42.2207}=20e^{4.22207}≈20*68.1743≈1363.486Wait, there's a discrepancy here. R_H(t) is higher than R_S(t) at t=42.2207, which suggests that the root is slightly higher than 42.2207.Wait, perhaps I need to do another iteration.Alternatively, perhaps I should accept that t≈42.22 months is the approximate solution, given the complexity of the calculations.So, summarizing:a) The time when the number of readers is equal is approximately t≈42.22 months.b) The reader count at this time is approximately 1363.7 thousand readers.But to be more precise, perhaps I can use a calculator to find a better approximation.Alternatively, I can use the fact that at t=42.22, R_H(t)=1363.92 and R_S(t)=1363.486, so the difference is about 0.434 thousand readers. To make them equal, we need to increase t slightly.The derivative of R_H(t) is R_H’(t)=10 + tAt t=42.22, R_H’(t)=10 +42.22=52.22The derivative of R_S(t) is R_S’(t)=2e^{0.1t}At t=42.22, R_S’(t)=2e^{4.222}≈2*68.1743≈136.3486So, the rate at which R_S(t) is growing is much higher than R_H(t). Thus, to make R_H(t)=R_S(t), we need to increase t slightly beyond 42.22.The difference in readers is ΔR=1363.92 -1363.486=0.434The rate difference is dR/dt= R_S’(t) - R_H’(t)=136.3486 -52.22≈84.1286Thus, the time needed to cover the difference is Δt=ΔR/(dR/dt)=0.434/84.1286≈0.00516 months.So, t≈42.22 +0.00516≈42.22516 months.Now, compute R_H(t) at t=42.22516:0.5*(42.22516)^2 +10*42.22516 +50=0.5*(1783.427) +422.2516 +50≈891.7135 +422.2516 +50≈1363.9651R_S(t)=20e^{0.1*42.22516}=20e^{4.222516}≈20*68.1743≈1363.486Wait, that's still a difference. Hmm, perhaps I need to iterate again.Alternatively, perhaps it's sufficient to state that t≈42.22 months, with R(t)≈1363.7 thousand readers.So, final answers:a) t≈42.22 monthsb) R(t)≈1363.7 thousand readersBut to be precise, perhaps I should round to two decimal places.Alternatively, if I use a calculator, I can find a more accurate value.But given the manual calculations, I'll go with t≈42.22 months and R(t)≈1363.7 thousand readers.

answer：First, I need to calculate the total cost of the educational trip. The program costs 1,500 per week, and the trip is 2 weeks long, so that's 3,000 for the program. Adding the airfare of 600 and travel insurance of 50, the total cost is 3,650.Jamie has already saved 1,200. To find out how much more is needed, I subtract the saved amount from the total cost: 3,650 minus 1,200 equals 2,450.Therefore, Jamie needs to save an additional 2,450 to cover the total cost of the trip.

answer：First, I need to calculate the total cost of the water polo balls. Each ball costs 15 and Coach Maria needs 8 of them. So, 15 multiplied by 8 equals 120.Next, I'll determine the total cost for the uniforms. Each uniform is 25, and she needs 7 uniforms. Multiplying 25 by 7 gives 175.Adding the cost of the balls and uniforms together, 120 plus 175 equals 295.Finally, I'll compare the total expenditure to the budget. Coach Maria has 500, and the total cost is 295. Since 295 is less than 500, she has enough money and will have 205 remaining.

answer：Alright, so I've got this problem about a priest wanting to redesign his church with some modern art. Specifically, he wants a spherical sculpture inside a paraboloid dome. The dome is given by the equation ( z = frac{1}{4}(x^2 + y^2) ), and the sphere has a radius of 2 meters. The sphere is centered at ( (0, 0, h) ), and it needs to be tangent to the dome at exactly one point. First, I need to figure out the height ( h ) where the center of the sphere should be placed. Then, once that's determined, I have to calculate the length of an LED strip around the sphere's equator. Starting with part 1: finding ( h ). I remember that for two surfaces to be tangent at a point, they must touch at that point and share a common tangent plane there. So, the sphere and the paraboloid must satisfy both the equation of the sphere and the equation of the paraboloid at the point of tangency, and their gradients (normals) must be equal at that point.The equation of the sphere is ( x^2 + y^2 + (z - h)^2 = r^2 ). Since the radius ( r ) is 2, that becomes ( x^2 + y^2 + (z - h)^2 = 4 ).The paraboloid is given by ( z = frac{1}{4}(x^2 + y^2) ). Let me denote ( x^2 + y^2 ) as ( s ) for simplicity, so ( z = frac{s}{4} ).At the point of tangency, both equations must hold. So substituting ( z ) from the paraboloid into the sphere's equation:( s + left( frac{s}{4} - h right)^2 = 4 ).Let me expand that:( s + left( frac{s^2}{16} - frac{h s}{2} + h^2 right) = 4 ).Combine like terms:( s + frac{s^2}{16} - frac{h s}{2} + h^2 = 4 ).Multiply through by 16 to eliminate the fraction:( 16s + s^2 - 8h s + 16h^2 = 64 ).Rearrange terms:( s^2 + (16 - 8h)s + 16h^2 - 64 = 0 ).This is a quadratic in terms of ( s ). For the sphere and paraboloid to be tangent, this equation must have exactly one solution for ( s ). That means the discriminant of this quadratic must be zero.The discriminant ( D ) of ( a s^2 + b s + c = 0 ) is ( D = b^2 - 4ac ).Here, ( a = 1 ), ( b = (16 - 8h) ), and ( c = 16h^2 - 64 ).So,( D = (16 - 8h)^2 - 4 * 1 * (16h^2 - 64) ).Let me compute that step by step.First, compute ( (16 - 8h)^2 ):( (16 - 8h)^2 = 256 - 256h + 64h^2 ).Then, compute ( 4 * (16h^2 - 64) ):( 4 * 16h^2 = 64h^2 ),( 4 * (-64) = -256 ),so altogether, it's ( 64h^2 - 256 ).Now, subtract the second result from the first:( D = (256 - 256h + 64h^2) - (64h^2 - 256) ).Simplify term by term:256 - 256h + 64h^2 - 64h^2 + 256.The ( 64h^2 ) terms cancel out.256 + 256 = 512.So, ( D = 512 - 256h ).For the quadratic to have exactly one solution, ( D = 0 ):( 512 - 256h = 0 ).Solving for ( h ):256h = 512,h = 512 / 256,h = 2.Wait, that seems straightforward. So the center of the sphere is at ( z = 2 ). But let me verify that because sometimes when dealing with quadratics, especially in geometry, it's easy to make a mistake.Let me plug ( h = 2 ) back into the equation.The sphere equation becomes ( x^2 + y^2 + (z - 2)^2 = 4 ).The paraboloid is ( z = frac{1}{4}(x^2 + y^2) ).Substituting ( z ) into the sphere equation:( x^2 + y^2 + left( frac{x^2 + y^2}{4} - 2 right)^2 = 4 ).Let ( s = x^2 + y^2 ), so:( s + left( frac{s}{4} - 2 right)^2 = 4 ).Compute ( left( frac{s}{4} - 2 right)^2 = frac{s^2}{16} - s + 4 ).So, the equation becomes:( s + frac{s^2}{16} - s + 4 = 4 ).Simplify:( frac{s^2}{16} + 0 + 4 = 4 ).Subtract 4 from both sides:( frac{s^2}{16} = 0 ).Which implies ( s^2 = 0 ), so ( s = 0 ).Therefore, the only solution is ( x = 0 ), ( y = 0 ), and ( z = 0 ). Wait, that can't be right because the sphere is centered at ( (0, 0, 2) ) with radius 2, so it would extend from ( z = 0 ) to ( z = 4 ). So, the point of tangency is at the origin? But the origin is the vertex of the paraboloid. Is the sphere tangent to the paraboloid only at the origin?Wait, that seems odd because the sphere is sitting above the origin, but if it's tangent at the origin, that would mean the sphere touches the paraboloid exactly at the lowest point. But the sphere's center is at ( z = 2 ), so the distance from the center to the origin is 2, which is equal to the radius. So, yes, the sphere touches the paraboloid at the origin and nowhere else.But is that the only point of contact? Let me visualize. The paraboloid is opening upwards, and the sphere is sitting inside it, touching at the origin. Since the sphere's radius is 2, and it's centered at ( z = 2 ), the bottom of the sphere is at ( z = 0 ), which is the vertex of the paraboloid. The sides of the sphere would extend outwards, but the paraboloid is curving upwards, so maybe they don't intersect elsewhere.Wait, but actually, when I solved the equation, I found that the only solution is at ( s = 0 ), which is the origin. So, that seems correct. Therefore, the sphere is tangent to the paraboloid only at the origin when ( h = 2 ).But wait, intuitively, if the sphere is centered higher up, wouldn't it touch the paraboloid somewhere else? Maybe I made a mistake in my substitution.Let me try another approach. Maybe using calculus to find where the distance from the sphere's center to the paraboloid is equal to the radius.The distance from the center ( (0, 0, h) ) to a point ( (x, y, z) ) on the paraboloid is ( sqrt{x^2 + y^2 + (z - h)^2} ). We want this distance to be equal to 2, the radius, and also, the point should lie on the paraboloid, so ( z = frac{1}{4}(x^2 + y^2) ).So, substituting ( z ) into the distance equation:( sqrt{x^2 + y^2 + left( frac{x^2 + y^2}{4} - h right)^2} = 2 ).Squaring both sides:( x^2 + y^2 + left( frac{x^2 + y^2}{4} - h right)^2 = 4 ).Let me let ( s = x^2 + y^2 ) again, so:( s + left( frac{s}{4} - h right)^2 = 4 ).Expanding the square:( s + frac{s^2}{16} - frac{h s}{2} + h^2 = 4 ).Multiply through by 16:( 16s + s^2 - 8h s + 16h^2 = 64 ).Rearranged:( s^2 + (16 - 8h)s + 16h^2 - 64 = 0 ).This is the same quadratic as before. So, discriminant must be zero for tangency:( D = (16 - 8h)^2 - 4(1)(16h^2 - 64) ).Which we found earlier to be ( 512 - 256h ). Setting this equal to zero gives ( h = 2 ).So, that seems consistent. Therefore, ( h = 2 ) meters.But just to make sure, let's think about the geometry. The sphere is centered at ( z = 2 ), radius 2, so it touches the origin. The paraboloid at the origin is just a point. As we move away from the origin, the paraboloid rises, and the sphere is also expanding. But because the sphere's center is at ( z = 2 ), and it's only radius 2, it can't reach much higher than ( z = 4 ). The paraboloid at ( z = 4 ) is ( x^2 + y^2 = 16 ), which is a circle of radius 4. The sphere at ( z = 4 ) is just a single point, so they don't intersect there. So, the only point of contact is at the origin.Therefore, I think ( h = 2 ) is correct.Moving on to part 2: calculating the length of the LED strip around the sphere's equator.The equator of the sphere is the circle where ( z = h ), because the sphere is centered at ( (0, 0, h) ). So, the equator is the intersection of the sphere with the plane ( z = h ).The sphere's equation is ( x^2 + y^2 + (z - h)^2 = 4 ). At ( z = h ), this simplifies to ( x^2 + y^2 = 4 ). So, the equator is a circle with radius 2 in the plane ( z = h ).Wait, but hold on, if the sphere is centered at ( (0, 0, h) ), then the equator is indeed the circle where ( z = h ), and the radius of that circle is equal to the sphere's radius, which is 2. Therefore, the circumference of the equator is ( 2pi r = 2pi * 2 = 4pi ) meters.But wait, let me think again. The sphere is in 3D space, so the equator is a great circle with radius equal to the sphere's radius. So, yes, circumference is ( 2pi r = 4pi ).But hold on, is the equator really at ( z = h )? Because in standard terms, the equator of a sphere is the circle that is equidistant from the top and bottom. But in this case, the sphere is centered at ( (0, 0, h) ), so the equator is indeed the plane cutting through the center, which is ( z = h ). So, that makes sense.Therefore, the length of the LED strip required is the circumference of this circle, which is ( 4pi ) meters.But just to double-check, let's compute it using the sphere's equation. At ( z = h ), the sphere equation becomes ( x^2 + y^2 + (h - h)^2 = 4 ), so ( x^2 + y^2 = 4 ). The circumference is ( 2pi sqrt{4} = 4pi ). Yep, that's correct.So, summarizing:1. The height ( h ) is 2 meters.2. The length of the LED strip is ( 4pi ) meters.**Final Answer**1. The height ( h ) is boxed{2} meters.2. The length of the LED strip is boxed{4pi} meters.