🧠:<think>Alright, so I've got this problem about Aric and the Convergence Crystal. It's got two parts, both involving some math. Let me try to tackle them one by one.Starting with the first part: calculating the resonant frequency ( f ) of the crystal. The formula given is:[ f = frac{1}{2pi} sqrt{frac{k_{text{magic}} + k_{text{muggle}}}{m}} ]Okay, so I need to plug in the values for ( k_{text{magic}} ), ( k_{text{muggle}} ), and ( m ). Let me write those down:- ( k_{text{magic}} = 16 ) N/m- ( k_{text{muggle}} = 9 ) N/m- ( m = 0.5 ) kgFirst, I should add the two spring constants together. So, ( 16 + 9 = 25 ). That gives me the combined spring constant.Next, I need to divide that sum by the mass ( m ). So, ( 25 ) divided by ( 0.5 ). Hmm, dividing by 0.5 is the same as multiplying by 2, right? So, ( 25 times 2 = 50 ). So, the numerator inside the square root becomes 50.Now, take the square root of 50. Let me think, ( sqrt{50} ) is equal to ( sqrt{25 times 2} ), which is ( 5sqrt{2} ). So, ( sqrt{50} approx 7.071 ) Hz. But I'll keep it as ( 5sqrt{2} ) for exactness.Then, the formula says to multiply this by ( frac{1}{2pi} ). So, ( f = frac{1}{2pi} times 5sqrt{2} ). Let me compute that.Calculating ( 5sqrt{2} ) first, which is approximately ( 5 times 1.4142 = 7.071 ). Then, ( frac{7.071}{2pi} ). Since ( pi ) is approximately 3.1416, ( 2pi ) is about 6.2832. So, ( 7.071 / 6.2832 ) is roughly 1.125 Hz.Wait, let me double-check that division. 7.071 divided by 6.2832. Let me compute it step by step:6.2832 goes into 7.071 once, since 6.2832 * 1 = 6.2832. Subtract that from 7.071, we get approximately 0.7878. Then, 6.2832 goes into 0.7878 about 0.125 times because 6.2832 * 0.125 is approximately 0.7854. So, adding that up, it's about 1.125 Hz.So, the resonant frequency ( f ) is approximately 1.125 Hz.But let me see if I can express it more precisely. Since ( sqrt{50} = 5sqrt{2} ), then ( f = frac{5sqrt{2}}{2pi} ). That's an exact expression, but if I want a decimal approximation, it's about 1.125 Hz.Wait, let me verify the calculation again because sometimes when dealing with square roots and divisions, it's easy to make a mistake.So, ( k_{text{magic}} + k_{text{muggle}} = 16 + 9 = 25 ). Then, ( 25 / 0.5 = 50 ). Square root of 50 is indeed ( 5sqrt{2} ). Then, ( 5sqrt{2} ) divided by ( 2pi ). So, ( 5sqrt{2} approx 5 times 1.4142 = 7.071 ). Then, ( 7.071 / (2pi) approx 7.071 / 6.2832 approx 1.125 ). Yeah, that seems correct.So, part 1 seems done. The resonant frequency is approximately 1.125 Hz.Moving on to part 2. This one is about a magical rune matrix ( A ), which is a 2x2 matrix with integer entries. The eigenvalues must be real and positive for the crystal to function properly, and the determinant of ( A ) is 5. I need to find a possible matrix ( A ) and its eigenvalues.Alright, so let's recall some linear algebra. For a 2x2 matrix, the eigenvalues can be found using the characteristic equation:[ lambda^2 - text{tr}(A)lambda + det(A) = 0 ]Where ( text{tr}(A) ) is the trace of the matrix (sum of diagonal elements) and ( det(A) ) is the determinant.Given that the determinant is 5, which is a prime number, and the matrix has integer entries, the determinant is the product of the eigenvalues. Since the eigenvalues must be real and positive, the trace (sum of eigenvalues) must also be positive.So, we need two positive real numbers whose product is 5 and whose sum is the trace of the matrix. Since 5 is prime, the possible pairs of positive integers whose product is 5 are (1,5). But wait, eigenvalues don't have to be integers, just real and positive. However, since the matrix has integer entries, the trace will be an integer, and the determinant is 5.But let me think. If the eigenvalues are real and positive, they could be irrational as well, but since the determinant is 5, which is an integer, and the trace is also an integer, the characteristic equation will have integer coefficients.But if the eigenvalues are irrational, their sum and product would still be integers. For example, ( lambda = frac{text{tr}(A) pm sqrt{text{tr}(A)^2 - 4 times 5}}{2} ). So, the discriminant must be a perfect square for the eigenvalues to be rational, but they can still be real even if the discriminant is not a perfect square.But since the problem doesn't specify that the eigenvalues need to be integers, just real and positive, we can have matrices where the eigenvalues are real and positive but not necessarily integers.However, the matrix ( A ) must have integer entries. So, perhaps the simplest way is to choose a diagonal matrix where the diagonal entries multiply to 5. Since 5 is prime, the possible integer pairs are (1,5) or (-1,-5), but since we need positive eigenvalues, we'll go with (1,5).So, a diagonal matrix with 1 and 5 on the diagonal would have determinant 5 and trace 6. The eigenvalues would be 1 and 5, which are real and positive.But maybe the problem expects a non-diagonal matrix? Let me see if that's possible.Alternatively, consider a matrix with trace 6 and determinant 5. The characteristic equation would be ( lambda^2 - 6lambda + 5 = 0 ), which factors as ( (lambda - 1)(lambda - 5) = 0 ), so eigenvalues 1 and 5.So, one such matrix is:[ A = begin{pmatrix} 1 & 0 0 & 5 end{pmatrix} ]But maybe a more interesting matrix is one that isn't diagonal. For example, a Jordan block or something else.Alternatively, consider a matrix with integer entries that isn't diagonal but still has eigenvalues 1 and 5. For example:[ A = begin{pmatrix} a & b c & d end{pmatrix} ]With ( a + d = 6 ) and ( ad - bc = 5 ).Let me try to find such integers ( a, b, c, d ).Let me pick ( a = 2 ), then ( d = 4 ) because ( 2 + 4 = 6 ). Then, ( ad = 8 ). So, ( bc = ad - 5 = 8 - 5 = 3 ). So, ( bc = 3 ). So, possible integer pairs for ( b ) and ( c ) are (1,3), (3,1), (-1,-3), (-3,-1).Let me choose ( b = 1 ) and ( c = 3 ). Then, the matrix is:[ A = begin{pmatrix} 2 & 1 3 & 4 end{pmatrix} ]Let me verify the determinant: ( (2)(4) - (1)(3) = 8 - 3 = 5 ). Correct. The trace is 6, so eigenvalues should be 1 and 5.Let me compute the eigenvalues to confirm.The characteristic equation is ( lambda^2 - 6lambda + 5 = 0 ), which factors as ( (lambda - 1)(lambda - 5) = 0 ). So, eigenvalues are 1 and 5, which are real and positive. Perfect.So, another possible matrix is:[ A = begin{pmatrix} 2 & 1 3 & 4 end{pmatrix} ]Alternatively, if I choose ( a = 3 ), then ( d = 3 ) because ( 3 + 3 = 6 ). Then, ( ad = 9 ). So, ( bc = 9 - 5 = 4 ). So, ( bc = 4 ). Possible pairs: (1,4), (2,2), (4,1), (-1,-4), etc.Let me choose ( b = 2 ), ( c = 2 ). Then, the matrix is:[ A = begin{pmatrix} 3 & 2 2 & 3 end{pmatrix} ]Determinant is ( 3*3 - 2*2 = 9 - 4 = 5 ). Trace is 6. Eigenvalues are 1 and 5, same as before.So, this is another possible matrix.Alternatively, if I choose ( a = 5 ), ( d = 1 ), then ( bc = 5*1 - 5 = 0 ). So, ( bc = 0 ). So, either ( b = 0 ) or ( c = 0 ). Then, the matrix would be diagonal or have a zero off-diagonal.For example, ( A = begin{pmatrix} 5 & 0 0 & 1 end{pmatrix} ), which is just a diagonal matrix again.So, there are multiple possibilities. The simplest is the diagonal matrix, but there are non-diagonal ones as well.So, to answer the question, I need to provide a possible matrix ( A ) and its eigenvalues.I think choosing a non-diagonal matrix would be more interesting, so I'll go with:[ A = begin{pmatrix} 2 & 1 3 & 4 end{pmatrix} ]Eigenvalues are 1 and 5.Alternatively, another matrix could be:[ A = begin{pmatrix} 3 & 2 2 & 3 end{pmatrix} ]With the same eigenvalues.Either would work. I think the first one is fine.So, summarizing:1. The resonant frequency ( f ) is approximately 1.125 Hz, or exactly ( frac{5sqrt{2}}{2pi} ) Hz.2. A possible matrix ( A ) is ( begin{pmatrix} 2 & 1 3 & 4 end{pmatrix} ) with eigenvalues 1 and 5.I think that's it. Let me just double-check the calculations to make sure I didn't make any mistakes.For part 1:- ( k_{text{magic}} + k_{text{muggle}} = 16 + 9 = 25 )- ( 25 / 0.5 = 50 )- ( sqrt{50} = 5sqrt{2} approx 7.071 )- ( 7.071 / (2pi) approx 1.125 ) HzLooks good.For part 2:- Determinant is 5, so eigenvalues multiply to 5.- Trace is 6, so eigenvalues add to 6.- The matrix ( begin{pmatrix} 2 & 1 3 & 4 end{pmatrix} ) has determinant 5 and trace 6, leading to eigenvalues 1 and 5.Yep, that's correct.Final Answer1. The resonant frequency is boxed{dfrac{5sqrt{2}}{2pi}} Hz.2. A possible matrix ( A ) is (begin{pmatrix} 2 & 1 3 & 4 end{pmatrix}) with eigenvalues boxed{1} and boxed{5}.</think>

🧠:<think>Alright, so I've got these two sub-problems to solve related to Ifan Williams's musical compositions. Let me take them one by one and think through each step carefully.Starting with the first problem: The student plays a sequence of 24 notes, each following the frequency function ( f(n) = 440 cdot 2^{(n-1)/12} ). I need to find the sum of all these frequencies. Hmm, okay. So, each note is determined by its position ( n ) in the 12-tone scale, starting from A4 which is 440 Hz. Wait, so the function is defined for each ( n ), but the student is playing 24 notes. Since the scale is 12-tone, does that mean the sequence wraps around after 12 notes? Or is it extending beyond the 12-tone scale? Let me think. The function ( f(n) ) is defined for any positive integer ( n ), so each subsequent note is the next semitone up. So, after A4 (n=1), it goes to A#4 (n=2), B4 (n=3), C5 (n=4), and so on. So, 24 notes would take us two full octaves above A4, right? Because each octave is 12 semitones.So, the sequence of frequencies is a geometric progression where each term is multiplied by ( 2^{1/12} ) to get the next term. The first term is ( f(1) = 440 cdot 2^{(1-1)/12} = 440 cdot 2^{0} = 440 ) Hz. The common ratio ( r ) is ( 2^{1/12} ).Since it's a geometric series, the sum of the first ( N ) terms is given by ( S_N = a_1 cdot frac{r^N - 1}{r - 1} ). Here, ( a_1 = 440 ), ( r = 2^{1/12} ), and ( N = 24 ). So, plugging in the numbers:( S_{24} = 440 cdot frac{(2^{1/12})^{24} - 1}{2^{1/12} - 1} ).Simplifying the exponent in the numerator: ( (2^{1/12})^{24} = 2^{24/12} = 2^2 = 4 ). So, the numerator becomes ( 4 - 1 = 3 ). Therefore, the sum is:( S_{24} = 440 cdot frac{3}{2^{1/12} - 1} ).Hmm, I wonder if this can be simplified further or if it's already in the simplest form. Let me think. The denominator is ( 2^{1/12} - 1 ), which is approximately 0.059463, but since the problem asks for the answer in simplest form, I think leaving it in terms of exponents is acceptable. So, I can write the sum as ( 440 cdot frac{3}{2^{1/12} - 1} ).Wait, but maybe there's a way to express this without the denominator. Let me see. Alternatively, since ( 2^{1/12} ) is the twelfth root of 2, which is approximately 1.059463. But unless the problem expects a numerical approximation, I should keep it exact. So, I think the expression ( 440 cdot frac{3}{2^{1/12} - 1} ) is the simplest exact form.Moving on to the second problem: The student plays the first 10 harmonics based on the fundamental frequency ( f(1) ). The harmonic frequencies are given by ( f_k = k cdot f(1) ) for ( k = 1 ) to ( 10 ). I need to find the average frequency of these 10 harmonics.First, let's find ( f(1) ). From the given function, ( f(1) = 440 cdot 2^{(1-1)/12} = 440 cdot 1 = 440 ) Hz. So, the fundamental frequency is 440 Hz.Then, the harmonics are ( f_k = 440k ) for ( k = 1 ) to ( 10 ). So, the frequencies are 440, 880, 1320, ..., up to 4400 Hz.To find the average frequency, I need to compute the mean of these 10 terms. The mean is the sum of the terms divided by the number of terms. So, first, let's find the sum of the first 10 harmonics.Since the harmonics form an arithmetic sequence where each term increases by 440 Hz, the sum ( S ) of the first ( n ) terms of an arithmetic sequence is given by ( S = frac{n}{2} cdot (a_1 + a_n) ). Here, ( n = 10 ), ( a_1 = 440 ), and ( a_{10} = 440 times 10 = 4400 ).So, the sum is ( S = frac{10}{2} cdot (440 + 4400) = 5 cdot 4840 = 24200 ) Hz.Then, the average frequency is ( frac{24200}{10} = 2420 ) Hz.Wait, that seems straightforward. Let me double-check. The first term is 440, the last term is 4400, the number of terms is 10. The average is indeed the average of the first and last term multiplied by the number of terms, but since we're taking the mean, it's just the average of the first and last term. So, ( frac{440 + 4400}{2} = frac{4840}{2} = 2420 ) Hz. Yep, that matches.So, the average frequency is 2420 Hz.Wait, hold on a second. Is that correct? Because the average of an arithmetic sequence is indeed the average of the first and last term. So, yes, that's correct. So, 2420 Hz is the mean frequency.Alright, so to recap:1. The sum of the frequencies of the 24 notes is ( 440 cdot frac{3}{2^{1/12} - 1} ).2. The average frequency of the first 10 harmonics is 2420 Hz.I think that's it. Let me just make sure I didn't make any calculation errors.For the first problem, the key was recognizing it's a geometric series with 24 terms, common ratio ( 2^{1/12} ), and first term 440. The sum formula applies, and simplifying the exponent gives 4 in the numerator, leading to 3 in the numerator after subtracting 1. So, the sum is ( 440 times 3 / (2^{1/12} - 1) ). That seems right.For the second problem, the harmonics are multiples of 440, so it's an arithmetic sequence with common difference 440. The average is the average of the first and last term, which is (440 + 4400)/2 = 2420. Yep, that's solid.I think I'm confident with these answers.Final Answer1. The sum of the frequencies is boxed{dfrac{1320}{2^{1/12} - 1}}.2. The average frequency of the first 10 harmonics is boxed{2420} Hz.</think>

🧠:<think>Alright, so I have this problem about fuel efficiency of two types of aircraft, Type A and Type B, over a 20-year period from 2000 to 2020. The officer wants to know when Type A first becomes more efficient than Type B and whether either type will reach 8 km/l by 2025. Hmm, okay, let me break this down step by step.First, let me write down the given functions for each type:For Type A:[ E_A(t) = 5 + 0.02t + sinleft(frac{pi t}{5}right) ]For Type B:[ E_B(t) = 4.5 + 0.03t + cosleft(frac{pi t}{4}right) ]Where ( t ) is the number of years since 2000. So, t ranges from 0 to 20 for the first part of the problem.Problem 1: Determine the year when Type A first exceeds Type B.I need to find the smallest t (where t is an integer from 0 to 20) such that ( E_A(t) > E_B(t) ).Let me set up the inequality:[ 5 + 0.02t + sinleft(frac{pi t}{5}right) > 4.5 + 0.03t + cosleft(frac{pi t}{4}right) ]Simplify this inequality:Subtract 4.5 from both sides:[ 0.5 + 0.02t + sinleft(frac{pi t}{5}right) > 0.03t + cosleft(frac{pi t}{4}right) ]Bring all terms to the left side:[ 0.5 + 0.02t - 0.03t + sinleft(frac{pi t}{5}right) - cosleft(frac{pi t}{4}right) > 0 ]Simplify the t terms:[ 0.5 - 0.01t + sinleft(frac{pi t}{5}right) - cosleft(frac{pi t}{4}right) > 0 ]So, the inequality becomes:[ 0.5 - 0.01t + sinleft(frac{pi t}{5}right) - cosleft(frac{pi t}{4}right) > 0 ]Let me denote this expression as ( f(t) ):[ f(t) = 0.5 - 0.01t + sinleft(frac{pi t}{5}right) - cosleft(frac{pi t}{4}right) ]We need to find the smallest integer t where ( f(t) > 0 ).Since t is an integer, I can compute ( f(t) ) for t = 0,1,2,... until I find the first t where f(t) > 0.Alternatively, maybe I can analyze the functions to see when this occurs.But since the functions involve sine and cosine, which are periodic, the difference might oscillate. So, perhaps the first time when Type A overtakes Type B is somewhere in the middle of the period.Alternatively, I can compute f(t) for each year from 2000 (t=0) onwards until I find when it becomes positive.Let me make a table of t from 0 to, say, 20, and compute f(t) each time.But before that, perhaps I can compute f(t) for t=0:t=0:f(0) = 0.5 - 0 + sin(0) - cos(0) = 0.5 - 1 = -0.5 < 0t=1:f(1) = 0.5 - 0.01 + sin(π/5) - cos(π/4)Compute each term:sin(π/5) ≈ sin(36°) ≈ 0.5878cos(π/4) ≈ √2/2 ≈ 0.7071So, f(1) ≈ 0.5 - 0.01 + 0.5878 - 0.7071 ≈ 0.5 - 0.01 = 0.49; 0.49 + 0.5878 = 1.0778; 1.0778 - 0.7071 ≈ 0.3707 > 0Wait, so at t=1, f(t) is already positive? That would mean in 2001, Type A is more efficient than Type B.But wait, let me double-check my calculations.Compute f(1):0.5 - 0.01*1 + sin(π*1/5) - cos(π*1/4)= 0.5 - 0.01 + sin(π/5) - cos(π/4)≈ 0.49 + 0.5878 - 0.7071≈ 0.49 + 0.5878 = 1.0778; 1.0778 - 0.7071 ≈ 0.3707Yes, that's correct. So, f(1) ≈ 0.3707 > 0.But wait, that seems too early. Let me check t=0:f(0) = 0.5 - 0 + 0 - 1 = -0.5 < 0So, between t=0 and t=1, f(t) crosses from negative to positive. Therefore, the first year when Type A exceeds Type B is 2001.But wait, the problem says "the year between 2000 and 2020 when the fuel efficiency of Type A aircraft first exceeds that of Type B aircraft."So, is it 2001? Hmm, but let me check t=1 in the original functions.Compute E_A(1) and E_B(1):E_A(1) = 5 + 0.02*1 + sin(π*1/5) ≈ 5 + 0.02 + 0.5878 ≈ 5.6078 km/lE_B(1) = 4.5 + 0.03*1 + cos(π*1/4) ≈ 4.5 + 0.03 + 0.7071 ≈ 5.2371 km/lSo, 5.6078 > 5.2371, so yes, Type A is more efficient in 2001.Wait, but is that correct? Because in 2000, t=0:E_A(0) = 5 + 0 + sin(0) = 5 km/lE_B(0) = 4.5 + 0 + cos(0) = 4.5 + 1 = 5.5 km/lSo, in 2000, Type B is more efficient. In 2001, Type A is more efficient.So, the first year when Type A exceeds Type B is 2001.But wait, let me check t=1 again:E_A(1) ≈ 5.6078E_B(1) ≈ 5.2371Yes, so 2001 is the first year.But wait, maybe I made a mistake in interpreting the functions. Let me check the functions again.For Type A: 5 + 0.02t + sin(πt/5)For Type B: 4.5 + 0.03t + cos(πt/4)Yes, that's correct.So, in 2000 (t=0):E_A = 5 + 0 + 0 = 5E_B = 4.5 + 0 + 1 = 5.5So, Type B is better.In 2001 (t=1):E_A ≈ 5 + 0.02 + 0.5878 ≈ 5.6078E_B ≈ 4.5 + 0.03 + 0.7071 ≈ 5.2371So, Type A is better.Therefore, the answer is 2001.Wait, but let me check t=0.5 to see if the crossing happens between t=0 and t=1.But since t is an integer (years), we only need to check integer values. So, the first integer t where f(t) > 0 is t=1, which is 2001.So, the answer to part 1 is 2001.Problem 2: Predict whether Type A or Type B will meet the 8 km/l requirement by 2025.So, we need to find if either E_A(t) or E_B(t) reaches or exceeds 8 km/l by t=25 (since 2025 is 25 years after 2000).First, let me check for Type A:We need to solve for t in E_A(t) ≥ 8.So,5 + 0.02t + sin(πt/5) ≥ 8Simplify:0.02t + sin(πt/5) ≥ 3Similarly, for Type B:4.5 + 0.03t + cos(πt/4) ≥ 8Simplify:0.03t + cos(πt/4) ≥ 3.5So, let's analyze both.Type A: 0.02t + sin(πt/5) ≥ 3Note that sin(πt/5) oscillates between -1 and 1. So, the maximum contribution from sin is +1, and minimum is -1.So, the maximum possible value of 0.02t + sin(πt/5) is 0.02t + 1.We need 0.02t + 1 ≥ 3 => 0.02t ≥ 2 => t ≥ 100.But 100 years is beyond our consideration (we only go up to t=25). So, even if sin(πt/5) is at its maximum, 0.02t + 1 would be 0.02*25 +1= 0.5 +1=1.5 <3. So, Type A cannot reach 8 km/l by 2025.Wait, that seems too quick. Let me check.Wait, E_A(t) =5 +0.02t + sin(πt/5). So, 5 +0.02t + sin(πt/5) ≥8 => 0.02t + sin(πt/5) ≥3.But 0.02t is linear, increasing by 0.02 each year, and sin(πt/5) oscillates between -1 and 1 with period 10 years.So, the maximum possible value of 0.02t + sin(πt/5) is 0.02t +1.We need 0.02t +1 ≥3 => 0.02t ≥2 => t≥100.So, even if sin is at maximum, it would take 100 years for Type A to reach 8 km/l. Since we only go up to 25 years, Type A won't reach 8.Type B: 0.03t + cos(πt/4) ≥3.5Similarly, cos(πt/4) oscillates between -1 and 1.Maximum contribution is +1, so maximum of 0.03t +1.Set 0.03t +1 ≥3.5 => 0.03t ≥2.5 => t≥2.5/0.03≈83.33 years.Again, beyond our timeframe.Wait, but maybe the oscillation could help? Let me think.Wait, cos(πt/4) has a period of 8 years. So, every 8 years, it completes a cycle.But since the coefficient of t is 0.03, which is small, the linear term is increasing, but the oscillation is between -1 and 1.So, even if cos(πt/4) is at its maximum, 0.03t +1 needs to be ≥3.5.Which, as above, requires t≥83.33, which is beyond 25.Wait, but maybe when cos(πt/4) is negative, it could subtract, but that would make it worse.Wait, but perhaps in some years, the combination of 0.03t and cos(πt/4) could reach 3.5 even if individually they don't.Wait, let's compute E_B(t) for t=25:E_B(25)=4.5 +0.03*25 + cos(π*25/4)Compute each term:0.03*25=0.75π*25/4≈19.635 radians. Let's compute cos(19.635).Since cos is periodic with period 2π≈6.283, so 19.635 /6.283≈3.125, so 3 full periods and 0.125*2π≈0.785 radians.So, cos(19.635)=cos(0.785)≈0.7071So, E_B(25)=4.5 +0.75 +0.7071≈5.9571 <8.So, even at t=25, E_B is only about 5.96 km/l, which is less than 8.Wait, but maybe earlier years have higher values?Wait, let's check t=20:E_B(20)=4.5 +0.03*20 +cos(π*20/4)=4.5+0.6 +cos(5π)=4.5+0.6 +(-1)=4.1 <8t=15:E_B(15)=4.5 +0.45 +cos(π*15/4)=4.95 +cos(3.75π)=cos(3.75π)=cos(π + 0.75π)= -cos(0.75π)= -(-√2/2)=√2/2≈0.7071So, E_B(15)=4.95 +0.7071≈5.6571 <8t=10:E_B(10)=4.5 +0.3 +cos(π*10/4)=4.8 +cos(2.5π)=cos(2.5π)=0So, E_B(10)=4.8 +0=4.8 <8t=5:E_B(5)=4.5 +0.15 +cos(π*5/4)=4.65 +cos(1.25π)=cos(1.25π)= -√2/2≈-0.7071So, E_B(5)=4.65 -0.7071≈3.9429 <8t=0:E_B(0)=4.5 +0 +cos(0)=4.5 +1=5.5 <8So, in all these years, E_B(t) is less than 8.Wait, but maybe at some point between t=25 and higher, but since the problem is up to 2025, which is t=25, and even at t=25, it's only ~5.96, which is still less than 8.Therefore, neither Type A nor Type B will reach 8 km/l by 2025.Wait, but let me double-check my calculations for Type B at t=25.E_B(25)=4.5 +0.03*25 +cos(π*25/4)0.03*25=0.75π*25/4≈19.635 radianscos(19.635)=cos(19.635 - 3*2π)=cos(19.635 -18.849)=cos(0.786)=≈0.7071So, E_B(25)=4.5+0.75+0.7071≈5.9571Yes, correct.So, neither type reaches 8 km/l by 2025.But wait, let me check if perhaps in some year between 2000 and 2025, the efficiency might peak higher due to the sine and cosine functions.For Type A, the maximum possible E_A(t) is 5 +0.02t +1=6 +0.02t.At t=25, that would be 6 +0.5=6.5 <8.Similarly, for Type B, the maximum E_B(t)=4.5 +0.03t +1=5.5 +0.03t.At t=25, that's 5.5 +0.75=6.25 <8.So, even at maximum, they don't reach 8.Therefore, neither Type A nor Type B will meet the 8 km/l requirement by 2025.Wait, but let me think again. Maybe I made a mistake in interpreting the functions.Wait, for Type A, E_A(t)=5 +0.02t +sin(πt/5). The maximum of sin is 1, so E_A_max=5 +0.02t +1=6 +0.02t.Similarly, for Type B, E_B_max=4.5 +0.03t +1=5.5 +0.03t.So, for Type A to reach 8:6 +0.02t ≥8 =>0.02t≥2 =>t≥100.For Type B:5.5 +0.03t ≥8 =>0.03t≥2.5 =>t≥83.33.So, both would need over 80 years, which is beyond 2025.Therefore, neither will meet the requirement by 2025.So, the answer to part 2 is that neither Type A nor Type B will meet the 8 km/l requirement by 2025.Wait, but let me check if perhaps the functions could reach 8 before t=25.Wait, for Type A, E_A(t)=5 +0.02t +sin(πt/5). Let's see when 5 +0.02t +1=6 +0.02t=8 =>0.02t=2 =>t=100.Similarly, Type B: 4.5 +0.03t +1=5.5 +0.03t=8 =>0.03t=2.5 =>t≈83.33.So, yes, both would need over 80 years.Therefore, neither will reach 8 km/l by 2025.So, summarizing:1. The first year Type A exceeds Type B is 2001.2. Neither Type A nor Type B will reach 8 km/l by 2025.But wait, let me check if perhaps in some year, the sine or cosine term could give a higher boost.Wait, for Type A, the sine function is sin(πt/5), which has a period of 10 years. So, every 10 years, it completes a cycle.Similarly, for Type B, the cosine function is cos(πt/4), period 8 years.So, perhaps in some years, the sine or cosine could be at their maximum, giving an extra boost.But as calculated, even with maximum boost, Type A would need 100 years, Type B 83 years.So, within 25 years, neither can reach 8.Therefore, the answers are:1. 20012. Neither will meet the requirement by 2025.Wait, but the problem says "at least one of the aircraft types must achieve an average fuel efficiency of 8 kilometers per liter by 2025." So, the answer is that neither will meet the requirement.But let me double-check for t=25:E_A(25)=5 +0.02*25 +sin(π*25/5)=5 +0.5 +sin(5π)=5.5 +0=5.5 <8E_B(25)=4.5 +0.03*25 +cos(π*25/4)=4.5 +0.75 +cos(19.635)=5.25 +0.7071≈5.9571 <8Yes, both are below 8.So, the conclusion is correct.</think>

🧠:<think>Okay, so I have this problem about a publishing agent who wants to turn a YouTuber's story into a book. The agent is using video analytics to predict the book's success. There are two parts to the problem. Let me try to tackle them one by one.Starting with part 1: Derive the expected potential book sales, E[S]. The model given is S = sum from i=1 to N of (a_i * R_i + b_i * E_i). So, S is a linear combination of the reach and engagement of each video. Given that R_i follows a Poisson distribution with mean λ_i, and E_i follows a normal distribution with mean μ_i and variance σ_i². I need to find the expectation of S.I remember that expectation is linear, so the expectation of a sum is the sum of expectations. So, E[S] = sum from i=1 to N of E[a_i * R_i + b_i * E_i]. Since expectation is linear, this can be split into sum from i=1 to N of (a_i * E[R_i] + b_i * E[E_i]). Now, for a Poisson distribution, the mean is λ_i, so E[R_i] = λ_i. For a normal distribution, the mean is μ_i, so E[E_i] = μ_i. Therefore, substituting these in, E[S] = sum from i=1 to N of (a_i * λ_i + b_i * μ_i). So, that should be the expected potential book sales. That seems straightforward.Moving on to part 2: The agent wants to maximize E[S] by adjusting the release strategy, which affects R_i and E_i, subject to a budget constraint. The constraint is sum from i=1 to N of (c_i * R_i + d_i * E_i) ≤ K, where c_i and d_i are cost coefficients, and K is the budget limit.So, we need to formulate this as an optimization problem. The objective function is E[S], which we already found as sum(a_i * λ_i + b_i * μ_i). But wait, in the optimization, are we adjusting R_i and E_i? Or are we adjusting something else?Wait, the problem says "adjusting the release strategy of videos to affect their reach and engagement." So, perhaps the agent can influence R_i and E_i by changing the release strategy, but subject to the cost constraint.But in the model, S is a function of R_i and E_i, which are random variables. However, in the expectation E[S], we have E[S] = sum(a_i * λ_i + b_i * μ_i). So, to maximize E[S], we need to maximize sum(a_i * λ_i + b_i * μ_i) subject to sum(c_i * R_i + d_i * E_i) ≤ K.But wait, R_i and E_i are random variables. How can we adjust them? Maybe the agent can influence the parameters of these distributions. For example, maybe λ_i and μ_i can be adjusted, which would affect the expected values.Alternatively, perhaps the agent can choose R_i and E_i, treating them as variables, but given that R_i is Poisson and E_i is normal, maybe it's more about setting their means.Wait, perhaps the problem is that the agent can choose the means λ_i and μ_i, subject to the cost constraint sum(c_i * λ_i + d_i * μ_i) ≤ K, because the cost is based on the expected values? Or is it based on the actual R_i and E_i?Wait, the constraint is sum(c_i * R_i + d_i * E_i) ≤ K. But R_i and E_i are random variables. So, if we're dealing with expectations, perhaps the constraint should be on the expectation as well? Or is it a probabilistic constraint?Hmm, the problem says "subject to the constraint sum(c_i * R_i + d_i * E_i) ≤ K". So, it's a hard constraint on the sum, but since R_i and E_i are random, this might be tricky. Maybe the agent wants to ensure that the expected cost is within the budget? Or perhaps the constraint is on the expectation.Wait, the problem says "adjusting the release strategy of videos to affect their reach and engagement", so maybe the agent can choose the parameters λ_i and μ_i, which in turn affect R_i and E_i. So, perhaps the constraint is on the expected cost: sum(c_i * E[R_i] + d_i * E[E_i]) ≤ K.But the problem doesn't specify whether the constraint is on the expectation or on the random variable itself. It just says "subject to the constraint sum(c_i * R_i + d_i * E_i) ≤ K". If we take it literally, it's a constraint on the random variable. But since the agent is trying to maximize the expectation, perhaps we can consider the expectation of the constraint? Or maybe it's a chance constraint, but the problem doesn't specify that.Alternatively, maybe the agent can choose R_i and E_i deterministically, but that contradicts the given distributions. So, perhaps the agent can set the means λ_i and μ_i, and the constraint is on the expected cost.Wait, let me think. If R_i is Poisson with mean λ_i, then E[R_i] = λ_i. Similarly, E[E_i] = μ_i. So, if the agent can choose λ_i and μ_i, then the constraint sum(c_i * R_i + d_i * E_i) ≤ K would have an expectation of sum(c_i * λ_i + d_i * μ_i) ≤ K.But the problem says "subject to the constraint sum(c_i * R_i + d_i * E_i) ≤ K". So, if we're dealing with expectations, then the constraint would be on the expectation. Otherwise, it's a stochastic constraint, which complicates things.Given that the problem is about maximizing E[S], which is the expectation, I think it's reasonable to assume that the constraint is also on the expectation. So, the optimization problem becomes:Maximize sum_{i=1}^N (a_i * λ_i + b_i * μ_i)Subject to sum_{i=1}^N (c_i * λ_i + d_i * μ_i) ≤ KAnd possibly, λ_i ≥ 0, μ_i ≥ 0, since they are means.So, this is a linear optimization problem in terms of variables λ_i and μ_i.To identify the conditions under which it has a unique solution, we can recall that linear programs have a unique solution if the objective function has a unique maximum over the feasible region, which is typically when the feasible region is a convex polyhedron and the objective function is not parallel to any of the edges of the feasible region.Alternatively, if the constraint matrix is full rank and the objective function is such that the maximum is attained at a unique vertex of the feasible region.But perhaps more specifically, for the problem to have a unique solution, the gradient of the objective function should not be orthogonal to any of the edges of the feasible region. In other words, the objective function should not be parallel to any of the constraints, ensuring that the maximum is achieved at a single point.Alternatively, if the feasible region is a convex set and the objective function is strictly increasing in some direction, then the maximum will be unique if the feasible region is bounded and the objective function is not constant over any edge.But in this case, since it's a linear program with variables λ_i and μ_i, the uniqueness of the solution depends on the uniqueness of the optimal solution in the feasible region.So, to have a unique solution, the system of equations formed by the active constraints at the optimum should have a unique solution, meaning that the number of active constraints equals the number of variables, and the corresponding matrix is invertible.In this case, the number of variables is 2N (each λ_i and μ_i), and the number of constraints is 1 (the budget constraint) plus the non-negativity constraints. So, unless 2N = 1, which is not the case, the system is underdetermined, so the solution may not be unique unless additional constraints make it so.Wait, but in reality, the agent might have more constraints, like λ_i ≥ 0 and μ_i ≥ 0, which are inequality constraints. So, the feasible region is defined by sum(c_i λ_i + d_i μ_i) ≤ K and λ_i ≥ 0, μ_i ≥ 0.In such a case, the feasible region is a convex polyhedron. The optimal solution will be at a vertex of this polyhedron. For the solution to be unique, the objective function must be such that it only touches the polyhedron at one vertex.This typically happens when the objective function is not parallel to any edge of the polyhedron. In other words, the gradient of the objective function (which is the vector of coefficients a_i and b_i) should not be orthogonal to any edge direction of the feasible region.Alternatively, if the feasible region is a simplex, and the objective function is such that it has a unique maximum, then the solution is unique.But perhaps more formally, in linear programming, a solution is unique if and only if the objective function's gradient is not orthogonal to any of the edges of the feasible region at the optimal solution.In our case, the feasible region is defined by the budget constraint and the non-negativity constraints. So, the vertices of the feasible region are points where some subset of variables are zero, and the budget constraint is tight.The optimal solution will be at a vertex where the objective function is maximized. For uniqueness, the objective function must not be parallel to any edge connected to that vertex.In terms of conditions, if the matrix formed by the coefficients of the active constraints (in this case, the budget constraint and some non-negativity constraints) has full rank, then the solution is unique.But since the budget constraint is a single equation, and the non-negativity constraints are inequalities, the active constraints at the optimal solution will include the budget constraint and possibly some non-negativity constraints.For the solution to be unique, the number of active constraints must equal the number of variables, but since we have 2N variables and only 1 budget constraint plus potentially up to 2N non-negativity constraints, it's possible that the solution is unique if the budget constraint is tight and the gradient of the objective function is such that it only intersects the budget constraint at one point within the feasible region.Alternatively, if all the coefficients in the objective function are positive, and the budget constraint is such that the optimal solution is at a single point where all variables are positive, then the solution is unique.Wait, but in our case, the objective function is sum(a_i λ_i + b_i μ_i). So, if a_i and b_i are all positive, then increasing λ_i and μ_i increases the objective function. However, the budget constraint limits how much we can increase them.So, the optimal solution would be to set as many λ_i and μ_i as possible to their maximum values given the budget. But since the budget is a single constraint, we might have to distribute the budget across all variables.Wait, but without more constraints, it's possible that multiple combinations of λ_i and μ_i could satisfy the budget constraint and yield the same maximum objective value.Wait, no. If the objective function is linear and the feasible region is convex, then the maximum is achieved at an extreme point (vertex). So, if the feasible region is such that the maximum is achieved at only one vertex, then the solution is unique.But in our case, the feasible region is defined by sum(c_i λ_i + d_i μ_i) ≤ K and λ_i ≥ 0, μ_i ≥ 0. So, the vertices are points where all but one variable are zero, and the remaining variable is set to K divided by its coefficient.Wait, no, that's only if we have a single constraint. But in reality, with multiple variables, the vertices are points where a subset of variables are zero, and the others are determined by the budget constraint.But in our case, since we have 2N variables, the number of possible vertices is huge. However, the optimal solution will be at a vertex where the objective function is maximized.For the solution to be unique, the objective function must be such that only one vertex gives the maximum value.This typically happens when the objective function's gradient is not parallel to any edge of the feasible region. In other words, the ratio of the coefficients in the objective function should not match the ratio of the coefficients in the budget constraint for any pair of variables.Wait, more formally, suppose we have two variables, say λ_1 and μ_1. If a_1 / c_1 = b_1 / d_1, then increasing λ_1 and μ_1 proportionally would not change the objective function per unit cost, leading to multiple solutions along that edge. Therefore, to have a unique solution, for all i and j, a_i / c_i ≠ b_j / d_j, or something like that.Wait, perhaps more accurately, if for any two variables, the ratio of their objective coefficients to their cost coefficients is the same, then there could be multiple solutions along the line where those two variables are adjusted together.Therefore, to ensure uniqueness, the ratios a_i / c_i must be distinct for all i, and similarly for b_i / d_i, but I'm not sure.Alternatively, if the objective function's gradient is not orthogonal to any of the edges of the feasible region, meaning that the direction of the objective function is not aligned with any edge, then the solution is unique.But perhaps a simpler condition is that the objective function is such that it has a unique maximum over the feasible region, which is a convex set. For linear functions over convex sets, the maximum is achieved at an extreme point, and if the function is not constant over any edge, then the maximum is unique.Wait, but in our case, the feasible region is a convex polyhedron, and the maximum of a linear function over a convex polyhedron is achieved at a vertex. If the function is not constant over any edge, then the maximum is unique.But in our case, the function is sum(a_i λ_i + b_i μ_i), and the edges of the feasible region are along the directions where one variable is increased while others are held constant, subject to the budget constraint.So, for the solution to be unique, the gradient of the objective function should not be orthogonal to any edge direction. That is, for any pair of variables, the ratio of their coefficients in the objective function should not equal the ratio of their coefficients in the budget constraint.Wait, let's formalize this. Suppose we have two variables, λ_i and λ_j. The edge direction would be along the line where λ_i and λ_j are adjusted while keeping other variables fixed. The direction vector would be proportional to (c_j, -c_i) for λ_i and λ_j, because increasing λ_i by Δ requires decreasing λ_j by (c_i / c_j) Δ to keep the budget constraint.Similarly, for μ_i and μ_j, the direction vector would be proportional to (d_j, -d_i).For the objective function, the gradient is (a_1, a_2, ..., a_N, b_1, b_2, ..., b_N).For the solution to be unique, the gradient should not be orthogonal to any edge direction. That is, for any i ≠ j, a_i / c_i ≠ a_j / c_j, and similarly for the μ variables, b_i / d_i ≠ b_j / d_j.Wait, actually, more precisely, for any two variables, say λ_i and λ_j, the ratio a_i / c_i should not equal a_j / c_j. Similarly, for μ_i and μ_j, b_i / d_i should not equal b_j / d_j. And also, for λ_i and μ_j, a_i / c_i should not equal b_j / d_j.Because if a_i / c_i = b_j / d_j, then increasing λ_i and decreasing μ_j (or vice versa) could keep the budget constraint the same while not changing the objective function, leading to multiple solutions.Therefore, the condition for uniqueness is that for all i ≠ j, a_i / c_i ≠ a_j / c_j, and for all i ≠ j, b_i / d_i ≠ b_j / d_j, and for all i, j, a_i / c_i ≠ b_j / d_j.In other words, all the ratios of objective coefficients to cost coefficients must be distinct across all variables.If this condition is met, then the optimal solution is unique because the objective function's gradient is not parallel to any edge of the feasible region, ensuring that the maximum is achieved at a single vertex.So, putting it all together, the optimization problem is:Maximize sum_{i=1}^N (a_i λ_i + b_i μ_i)Subject to:sum_{i=1}^N (c_i λ_i + d_i μ_i) ≤ Kλ_i ≥ 0, μ_i ≥ 0 for all iAnd the solution is unique if and only if all the ratios a_i / c_i and b_i / d_i are distinct across all i, and no a_i / c_i equals any b_j / d_j.So, that's the condition for uniqueness.Wait, but let me double-check. Suppose we have two variables, λ_1 and μ_1. If a_1 / c_1 = b_1 / d_1, then increasing λ_1 and decreasing μ_1 proportionally could keep the budget constraint the same while not changing the objective function, leading to multiple solutions. So, yes, that ratio must be unique across all variables.Therefore, the conditions are that all a_i / c_i are distinct, all b_i / d_i are distinct, and no a_i / c_i equals any b_j / d_j.So, summarizing:The optimization problem is a linear program with variables λ_i and μ_i, maximizing the expected sales subject to the budget constraint and non-negativity constraints. The solution is unique if all the ratios of objective coefficients to cost coefficients are distinct across all variables.I think that's it.</think>