🧠:<think>Okay, so I have this problem about global military expenditures and their impact on international relations. The researcher is looking at a differential equation to model how military spending changes over time. The equation given is:[ frac{dM}{dt} = k M(t) - c M(t)^2 ]where ( k ) and ( c ) are positive constants, and ( M(0) = M_0 ). First, I need to solve this differential equation to find ( M(t) ). Hmm, this looks like a logistic growth model. I remember that the logistic equation is of the form:[ frac{dP}{dt} = r P - s P^2 ]where ( r ) is the growth rate and ( s ) is related to the carrying capacity. So, in this case, ( k ) would be like the growth rate, and ( c ) would be related to the carrying capacity. To solve this, I think I can use separation of variables. Let me rewrite the equation:[ frac{dM}{dt} = M(k - c M) ]So, separating variables, I get:[ frac{dM}{M(k - c M)} = dt ]Now, I need to integrate both sides. The left side looks like it can be integrated using partial fractions. Let me set it up:Let me denote:[ frac{1}{M(k - c M)} = frac{A}{M} + frac{B}{k - c M} ]Multiplying both sides by ( M(k - c M) ), I get:[ 1 = A(k - c M) + B M ]Expanding:[ 1 = A k - A c M + B M ]Grouping like terms:[ 1 = A k + ( - A c + B ) M ]Since this must hold for all ( M ), the coefficients of like terms must be equal on both sides. So, the coefficient of ( M ) must be zero, and the constant term must be 1.So,1. Coefficient of ( M ): ( - A c + B = 0 ) => ( B = A c )2. Constant term: ( A k = 1 ) => ( A = frac{1}{k} )Therefore, ( B = frac{c}{k} )So, the partial fractions decomposition is:[ frac{1}{M(k - c M)} = frac{1}{k M} + frac{c}{k(k - c M)} ]Therefore, the integral becomes:[ int left( frac{1}{k M} + frac{c}{k(k - c M)} right) dM = int dt ]Let me compute the left integral term by term.First term:[ int frac{1}{k M} dM = frac{1}{k} ln |M| + C_1 ]Second term:Let me make a substitution for the second integral. Let ( u = k - c M ), then ( du = -c dM ), so ( dM = -frac{1}{c} du )So,[ int frac{c}{k(k - c M)} dM = int frac{c}{k u} cdot left( -frac{1}{c} right) du = -frac{1}{k} int frac{1}{u} du = -frac{1}{k} ln |u| + C_2 = -frac{1}{k} ln |k - c M| + C_2 ]Putting it all together, the left integral is:[ frac{1}{k} ln |M| - frac{1}{k} ln |k - c M| + C ]Where ( C = C_1 + C_2 ). So, combining the logs:[ frac{1}{k} ln left| frac{M}{k - c M} right| = t + C ]Exponentiating both sides:[ left| frac{M}{k - c M} right| = e^{k(t + C)} = e^{k t} cdot e^{k C} ]Let me denote ( e^{k C} ) as another constant, say ( C' ). Since the absolute value can be absorbed into the constant, we can write:[ frac{M}{k - c M} = C' e^{k t} ]Now, solving for ( M ):Multiply both sides by ( (k - c M) ):[ M = C' e^{k t} (k - c M) ]Expanding:[ M = C' k e^{k t} - C' c e^{k t} M ]Bring the term with ( M ) to the left:[ M + C' c e^{k t} M = C' k e^{k t} ]Factor out ( M ):[ M (1 + C' c e^{k t}) = C' k e^{k t} ]Therefore,[ M = frac{C' k e^{k t}}{1 + C' c e^{k t}} ]Now, let's apply the initial condition ( M(0) = M_0 ). At ( t = 0 ):[ M(0) = frac{C' k e^{0}}{1 + C' c e^{0}} = frac{C' k}{1 + C' c} = M_0 ]Solving for ( C' ):Let me denote ( C' = D ) for simplicity.So,[ frac{D k}{1 + D c} = M_0 ]Multiply both sides by ( 1 + D c ):[ D k = M_0 (1 + D c) ]Expand:[ D k = M_0 + M_0 D c ]Bring all terms with ( D ) to the left:[ D k - M_0 D c = M_0 ]Factor out ( D ):[ D (k - M_0 c) = M_0 ]Therefore,[ D = frac{M_0}{k - M_0 c} ]So, substituting back into the expression for ( M(t) ):[ M(t) = frac{ left( frac{M_0}{k - M_0 c} right) k e^{k t} }{1 + left( frac{M_0}{k - M_0 c} right) c e^{k t} } ]Simplify numerator and denominator:Numerator:[ frac{M_0 k e^{k t}}{k - M_0 c} ]Denominator:[ 1 + frac{M_0 c e^{k t}}{k - M_0 c} = frac{(k - M_0 c) + M_0 c e^{k t}}{k - M_0 c} = frac{k - M_0 c + M_0 c e^{k t}}{k - M_0 c} ]Therefore, ( M(t) ) is:[ M(t) = frac{ frac{M_0 k e^{k t}}{k - M_0 c} }{ frac{k - M_0 c + M_0 c e^{k t}}{k - M_0 c} } = frac{M_0 k e^{k t}}{k - M_0 c + M_0 c e^{k t}} ]We can factor out ( k ) in the denominator:Wait, actually, let me factor ( e^{k t} ) in the denominator:[ k - M_0 c + M_0 c e^{k t} = k - M_0 c (1 - e^{k t}) ]But maybe it's better to write it as:[ M(t) = frac{M_0 k e^{k t}}{k + M_0 c (e^{k t} - 1)} ]Alternatively, we can factor ( e^{k t} ) in the numerator and denominator:[ M(t) = frac{M_0 k e^{k t}}{k - M_0 c + M_0 c e^{k t}} = frac{M_0 k}{(k - M_0 c) e^{-k t} + M_0 c} ]But perhaps the first form is simpler. Let me check if this makes sense. When ( t = 0 ), ( M(0) = frac{M_0 k}{k - M_0 c + M_0 c} = frac{M_0 k}{k} = M_0 ), which is correct.As ( t ) approaches infinity, ( e^{k t} ) dominates, so:[ M(t) approx frac{M_0 k e^{k t}}{M_0 c e^{k t}} = frac{k}{c} ]So, the carrying capacity is ( frac{k}{c} ), which is consistent with the logistic model.Okay, so that seems correct.So, the solution to part 1 is:[ M(t) = frac{M_0 k e^{k t}}{k - M_0 c + M_0 c e^{k t}} ]Alternatively, it can be written as:[ M(t) = frac{M_0}{(M_0 / k) + (1 - M_0 / k) e^{-k t}} ]But the first expression is fine.Now, moving on to part 2. The researcher believes that non-interventionist policies would reduce global military expenditure by ( p ) percent annually. So, I need to modify the differential equation to reflect this belief.Hmm, reducing by ( p ) percent annually. So, that would mean that the growth rate ( k ) is reduced by ( p ) percent each year. Alternatively, perhaps the term ( k M(t) ) is reduced by ( p ) percent.Wait, let me think. If the expenditure is reduced by ( p ) percent annually, that could mean that the rate of change is decreased by ( p ) percent. So, perhaps the differential equation becomes:[ frac{dM}{dt} = (k - p) M(t) - c M(t)^2 ]But wait, ( p ) is a percentage, so if it's a reduction of ( p ) percent, then the new growth rate would be ( k (1 - p/100) ). So, perhaps:[ frac{dM}{dt} = k (1 - frac{p}{100}) M(t) - c M(t)^2 ]Alternatively, if the reduction is applied to the entire rate, maybe it's subtracting ( p % ) of ( M(t) ) each year. So, the differential equation would have an additional term subtracting ( frac{p}{100} M(t) ). So:[ frac{dM}{dt} = k M(t) - c M(t)^2 - frac{p}{100} M(t) ]Which can be written as:[ frac{dM}{dt} = (k - frac{p}{100}) M(t) - c M(t)^2 ]That seems reasonable. So, effectively, the growth rate ( k ) is decreased by ( frac{p}{100} ). So, the modified differential equation is:[ frac{dM}{dt} = (k - frac{p}{100}) M(t) - c M(t)^2 ]Alternatively, if ( p ) is given as a decimal, say ( p = 0.05 ) for 5%, then it's just ( k - p ). But since the problem says ( p ) percent, I think it's safer to write it as ( k - frac{p}{100} ).So, the modified equation is:[ frac{dM}{dt} = (k - frac{p}{100}) M(t) - c M(t)^2 ]Now, we need to solve this differential equation with the same initial condition ( M(0) = M_0 ).This is again a logistic equation, similar to part 1, but with a modified growth rate ( k' = k - frac{p}{100} ). So, if ( k' ) is positive, the solution will be similar, but if ( k' ) becomes negative, the behavior will change.Assuming ( k - frac{p}{100} > 0 ), which would mean that the growth rate is still positive, just reduced. If ( k - frac{p}{100} ) is negative, then the model would predict decay towards zero.But let's proceed assuming ( k' = k - frac{p}{100} ) is positive, so the solution will be similar to part 1.So, the differential equation is:[ frac{dM}{dt} = k' M(t) - c M(t)^2 ]with ( k' = k - frac{p}{100} ).So, following the same steps as in part 1, we can solve this.Rewrite the equation:[ frac{dM}{dt} = M(k' - c M) ]Separate variables:[ frac{dM}{M(k' - c M)} = dt ]Integrate both sides. Using partial fractions again:[ frac{1}{M(k' - c M)} = frac{A}{M} + frac{B}{k' - c M} ]Multiply both sides by ( M(k' - c M) ):[ 1 = A(k' - c M) + B M ]Expand:[ 1 = A k' - A c M + B M ]Group terms:[ 1 = A k' + ( - A c + B ) M ]So, coefficients must satisfy:1. ( A k' = 1 ) => ( A = frac{1}{k'} )2. ( - A c + B = 0 ) => ( B = A c = frac{c}{k'} )Therefore, partial fractions decomposition is:[ frac{1}{M(k' - c M)} = frac{1}{k' M} + frac{c}{k'(k' - c M)} ]Integrate both sides:Left side:[ int left( frac{1}{k' M} + frac{c}{k'(k' - c M)} right) dM = int dt ]First integral:[ frac{1}{k'} int frac{1}{M} dM = frac{1}{k'} ln |M| + C_1 ]Second integral:Let ( u = k' - c M ), then ( du = -c dM ), so ( dM = -frac{1}{c} du )Thus,[ frac{c}{k'} int frac{1}{u} cdot left( -frac{1}{c} right) du = -frac{1}{k'} int frac{1}{u} du = -frac{1}{k'} ln |u| + C_2 ]So, the left integral becomes:[ frac{1}{k'} ln |M| - frac{1}{k'} ln |k' - c M| + C ]Combine logs:[ frac{1}{k'} ln left| frac{M}{k' - c M} right| = t + C ]Exponentiate both sides:[ left| frac{M}{k' - c M} right| = e^{k' (t + C)} = e^{k' t} cdot e^{k' C} ]Let ( e^{k' C} = C' ), so:[ frac{M}{k' - c M} = C' e^{k' t} ]Solve for ( M ):Multiply both sides by ( (k' - c M) ):[ M = C' e^{k' t} (k' - c M) ]Expand:[ M = C' k' e^{k' t} - C' c e^{k' t} M ]Bring ( M ) terms to the left:[ M + C' c e^{k' t} M = C' k' e^{k' t} ]Factor out ( M ):[ M (1 + C' c e^{k' t}) = C' k' e^{k' t} ]Therefore,[ M = frac{C' k' e^{k' t}}{1 + C' c e^{k' t}} ]Apply initial condition ( M(0) = M_0 ):At ( t = 0 ):[ M(0) = frac{C' k'}{1 + C' c} = M_0 ]Solve for ( C' ):Let ( C' = D ):[ frac{D k'}{1 + D c} = M_0 ]Multiply both sides by ( 1 + D c ):[ D k' = M_0 (1 + D c) ]Expand:[ D k' = M_0 + M_0 D c ]Bring terms with ( D ) to the left:[ D k' - M_0 D c = M_0 ]Factor:[ D (k' - M_0 c) = M_0 ]Thus,[ D = frac{M_0}{k' - M_0 c} ]Substitute back into ( M(t) ):[ M(t) = frac{ left( frac{M_0}{k' - M_0 c} right) k' e^{k' t} }{1 + left( frac{M_0}{k' - M_0 c} right) c e^{k' t} } ]Simplify numerator and denominator:Numerator:[ frac{M_0 k' e^{k' t}}{k' - M_0 c} ]Denominator:[ 1 + frac{M_0 c e^{k' t}}{k' - M_0 c} = frac{(k' - M_0 c) + M_0 c e^{k' t}}{k' - M_0 c} ]Thus,[ M(t) = frac{M_0 k' e^{k' t}}{k' - M_0 c + M_0 c e^{k' t}} ]But remember that ( k' = k - frac{p}{100} ). So, substituting back:[ M(t) = frac{M_0 (k - frac{p}{100}) e^{(k - frac{p}{100}) t}}{(k - frac{p}{100}) - M_0 c + M_0 c e^{(k - frac{p}{100}) t}} ]Alternatively, we can write this as:[ M(t) = frac{M_0 k' e^{k' t}}{k' - M_0 c + M_0 c e^{k' t}} ]where ( k' = k - frac{p}{100} ).Let me check the initial condition again. At ( t = 0 ):[ M(0) = frac{M_0 k'}{k' - M_0 c + M_0 c} = frac{M_0 k'}{k'} = M_0 ]Good, that works.Also, as ( t ) approaches infinity, if ( k' > 0 ), then ( e^{k' t} ) dominates, so:[ M(t) approx frac{M_0 k' e^{k' t}}{M_0 c e^{k' t}} = frac{k'}{c} ]So, the carrying capacity is now ( frac{k'}{c} = frac{k - frac{p}{100}}{c} ), which is lower than before, as expected due to the reduced growth rate.If ( k' ) becomes negative, then the exponential term ( e^{k' t} ) would decay to zero, and ( M(t) ) would approach zero. So, if ( k - frac{p}{100} < 0 ), the model predicts that military expenditure would decay to zero over time.Therefore, the solution under non-interventionist policies is:[ M(t) = frac{M_0 (k - frac{p}{100}) e^{(k - frac{p}{100}) t}}{(k - frac{p}{100}) - M_0 c + M_0 c e^{(k - frac{p}{100}) t}} ]Alternatively, simplifying the expression:[ M(t) = frac{M_0 k' e^{k' t}}{k' - M_0 c + M_0 c e^{k' t}} ]where ( k' = k - frac{p}{100} ).So, summarizing:1. The original solution is ( M(t) = frac{M_0 k e^{k t}}{k - M_0 c + M_0 c e^{k t}} )2. Under non-interventionist policies, the solution is ( M(t) = frac{M_0 (k - frac{p}{100}) e^{(k - frac{p}{100}) t}}{(k - frac{p}{100}) - M_0 c + M_0 c e^{(k - frac{p}{100}) t}} )I think that's it. Let me just double-check the steps. In part 2, I modified the growth rate by subtracting ( p % ), which seems reasonable. The integration steps are the same as part 1, just with ( k' ) instead of ( k ). So, the solutions should be correct.</think>

🧠:<think>Alright, so I have two problems to solve here about these two rock bands from the 70s. Let me tackle them one by one.Starting with Band A. They released 8 albums in the 70s with a total of 96 songs. I need to find the mean and standard deviation of the number of songs per album, assuming it's normally distributed. The key piece of information given is that the probability of an album having between 10 and 14 songs is approximately 68.27%. Hmm, that number sounds familiar. I think 68.27% is the probability that a normally distributed variable lies within one standard deviation of the mean. So, that would mean that the interval from 10 to 14 songs covers one standard deviation on either side of the mean.Let me write that down. If the probability is 68.27%, then the interval 10 to 14 is one standard deviation around the mean. So, the mean should be the midpoint of 10 and 14. The midpoint is (10 + 14)/2 = 12. So, μ = 12. That makes sense because in a normal distribution, the mean is the center.Now, the standard deviation. Since 10 to 14 is one standard deviation, the distance from the mean to either end is σ. So, from 12 to 14 is 2, which should be equal to σ. Wait, no. If the interval is 10 to 14, that's a total width of 4. Since it's one standard deviation on each side, the total width is 2σ. So, 2σ = 4, which means σ = 2. Let me double-check that. If μ is 12 and σ is 2, then 12 - 2 = 10 and 12 + 2 = 14. Yep, that fits. So, the mean is 12 and the standard deviation is 2.Wait, but let me think again. The total number of songs is 96 across 8 albums. So, the average number of songs per album is 96/8 = 12. That's consistent with the mean we just found. So, that's a good check. So, Band A has μ = 12 and σ = 2.Moving on to Band B. They released 6 albums with a total of 78 songs. The number of songs per album forms an arithmetic sequence. The first album has 9 songs, and I need to find the number of songs on the last album and the common difference.Okay, arithmetic sequence. So, the number of songs on each album increases by a common difference each time. The total number of songs is the sum of the arithmetic sequence. The formula for the sum of an arithmetic sequence is S_n = n/2 * (2a + (n-1)d), where S_n is the sum, n is the number of terms, a is the first term, and d is the common difference.Given that, S_6 = 78, n = 6, a = 9. Plugging in the values: 78 = 6/2 * (2*9 + (6-1)d). Simplify that: 78 = 3*(18 + 5d). Divide both sides by 3: 26 = 18 + 5d. Subtract 18: 8 = 5d. So, d = 8/5 = 1.6. Hmm, that's a fractional number of songs. That doesn't make much sense because you can't have a fraction of a song on an album. Maybe I made a mistake.Wait, let me check my calculations again. S_6 = 6/2*(2*9 + 5d) = 3*(18 + 5d). So, 3*(18 + 5d) = 78. Divide both sides by 3: 18 + 5d = 26. Subtract 18: 5d = 8. So, d = 8/5 = 1.6. Yeah, same result. Hmm, maybe the common difference is 1.6, but that seems odd. Alternatively, maybe I misapplied the formula.Wait, another formula for the sum is S_n = n*(a1 + an)/2, where an is the nth term. Maybe that's easier. So, 78 = 6*(9 + a6)/2. Simplify: 78 = 3*(9 + a6). Divide both sides by 3: 26 = 9 + a6. So, a6 = 26 - 9 = 17. So, the last album has 17 songs.Now, to find the common difference d. Since it's an arithmetic sequence, the nth term is a1 + (n-1)d. So, a6 = 9 + 5d = 17. So, 5d = 17 - 9 = 8. Thus, d = 8/5 = 1.6 again. So, same result. So, the common difference is 1.6, which is 8/5. That seems correct mathematically, but in reality, you can't have a fraction of a song. Maybe the problem allows for it, or perhaps I misread the problem.Wait, the problem says the number of songs forms an arithmetic sequence, but it doesn't specify that the number has to be an integer. So, maybe it's possible. Alternatively, maybe I made a mistake in interpreting the problem. Let me check the total number of songs again. 6 albums, total 78. First album 9, last album 17, so the average per album is (9 + 17)/2 = 13, and 13*6 = 78. That checks out. So, the last album has 17 songs, and the common difference is 1.6.But 1.6 is 8/5, which is 1 and 3/5. So, each album increases by 1.6 songs. That seems unusual, but mathematically, it's correct. Maybe in the context of the problem, it's acceptable. Alternatively, perhaps the problem expects integer values, so maybe I need to reconsider.Wait, if the common difference has to be an integer, then maybe my initial approach is wrong. Let me see. If d is an integer, then 5d = 8, which would require d = 1.6, which isn't an integer. So, that's not possible. Therefore, maybe the problem allows for fractional songs, or perhaps I made a mistake in the setup.Wait, another thought: maybe the number of songs is an integer, so the common difference must be such that each term is an integer. So, starting at 9, adding d each time, and ending at 17. So, 9, 9+d, 9+2d, ..., 9+5d=17. So, 5d=8, d=1.6. So, unless d is a fraction, it's not possible. So, maybe the problem allows for fractional songs, or perhaps I misread the total number of songs.Wait, the total is 78. Let me check: 9 + (9+d) + (9+2d) + (9+3d) + (9+4d) + (9+5d) = 6*9 + (0+1+2+3+4+5)d = 54 + 15d. So, 54 + 15d = 78. Therefore, 15d = 24, so d = 24/15 = 8/5 = 1.6. Yep, same result. So, unless the problem allows for fractional songs, which is unusual, but perhaps in the context of an arithmetic sequence, it's acceptable.Alternatively, maybe the problem expects the common difference to be a fraction, so 1.6 is the answer. So, the last album has 17 songs, and the common difference is 1.6. So, I think that's the answer.Wait, but let me think again. If the common difference is 1.6, then the number of songs on each album would be: 9, 10.6, 12.2, 13.8, 15.4, 17. So, those are the number of songs. But you can't have a fraction of a song. So, perhaps the problem is designed in a way that the common difference is a fraction, or maybe I misread the total number of songs.Wait, the total is 78. Let me check: 9 + 10.6 + 12.2 + 13.8 + 15.4 + 17 = let's add them up. 9 + 17 = 26, 10.6 + 15.4 = 26, 12.2 + 13.8 = 26. So, 26*3 = 78. Yep, that's correct. So, even though the number of songs per album is fractional, the total adds up correctly. So, maybe the problem is designed that way, allowing for fractional songs in the arithmetic sequence.Alternatively, perhaps the problem expects the common difference to be an integer, and I need to adjust. But given the math, it's 1.6. So, I think that's the answer.So, to recap:For Band A: μ = 12, σ = 2.For Band B: Last album has 17 songs, common difference is 1.6.I think that's it.</think>

🧠:<think>Alright, so I have this problem about Elizabeth and her positivity club. It's divided into two parts. Let me tackle them one by one.Starting with part 1: The number of participants each week follows a quadratic sequence, P(n) = an² + bn + c. We know the number of participants for the first three weeks: 15, 22, and 33. I need to find the constants a, b, and c.Okay, so since it's a quadratic sequence, the formula is P(n) = an² + bn + c. We have three data points, so we can set up three equations and solve for a, b, and c.Let me write down the equations based on the given data:For week 1 (n=1): P(1) = a(1)² + b(1) + c = a + b + c = 15.For week 2 (n=2): P(2) = a(2)² + b(2) + c = 4a + 2b + c = 22.For week 3 (n=3): P(3) = a(3)² + b(3) + c = 9a + 3b + c = 33.So now we have a system of three equations:1) a + b + c = 152) 4a + 2b + c = 223) 9a + 3b + c = 33I need to solve this system. Let me subtract equation 1 from equation 2 to eliminate c:Equation 2 - Equation 1: (4a + 2b + c) - (a + b + c) = 22 - 15That simplifies to 3a + b = 7. Let's call this equation 4.Similarly, subtract equation 2 from equation 3:Equation 3 - Equation 2: (9a + 3b + c) - (4a + 2b + c) = 33 - 22Which simplifies to 5a + b = 11. Let's call this equation 5.Now, we have two equations:4) 3a + b = 75) 5a + b = 11Subtract equation 4 from equation 5:(5a + b) - (3a + b) = 11 - 7This gives 2a = 4, so a = 2.Now plug a = 2 into equation 4:3(2) + b = 7 => 6 + b = 7 => b = 1.Now, substitute a = 2 and b = 1 into equation 1:2 + 1 + c = 15 => 3 + c = 15 => c = 12.So, the quadratic function is P(n) = 2n² + n + 12.Let me double-check with the given data:For n=1: 2(1) + 1 + 12 = 15. Correct.For n=2: 2(4) + 2 + 12 = 8 + 2 + 12 = 22. Correct.For n=3: 2(9) + 3 + 12 = 18 + 3 + 12 = 33. Correct.Great, so part 1 is solved. a=2, b=1, c=12.Moving on to part 2: The positivity level Q(t) is modeled by Q(t) = A sin(Bt + C) + D. We know that the positivity hits a peak of 90 at t=4 and a minimum of 10 at t=9.We need to find A, B, C, D.First, let's recall that the general sine function is Q(t) = A sin(Bt + C) + D. The amplitude is A, the period is 2π/B, the phase shift is -C/B, and the vertical shift is D.Given that the maximum is 90 and the minimum is 10, we can find A and D.The amplitude A is half the difference between the maximum and minimum values. So:A = (Max - Min)/2 = (90 - 10)/2 = 80/2 = 40.The vertical shift D is the average of the maximum and minimum:D = (Max + Min)/2 = (90 + 10)/2 = 100/2 = 50.So, A=40 and D=50.Now, the function becomes Q(t) = 40 sin(Bt + C) + 50.Next, we need to find B and C.We know that the maximum occurs at t=4 and the minimum at t=9.In a sine function, the maximum occurs at π/2 and the minimum at 3π/2 within one period.The time between a maximum and the next minimum is half a period. So, the time between t=4 and t=9 is 5 weeks, which is half the period.Therefore, the period T is 10 weeks.Since period T = 2π/B, we can solve for B:2π/B = 10 => B = 2π/10 = π/5.So, B=π/5.Now, we can write the function as Q(t) = 40 sin((π/5)t + C) + 50.We need to find C. To do this, we can use one of the given points. Let's use the maximum at t=4.At t=4, Q(4)=90.So, 90 = 40 sin((π/5)(4) + C) + 50.Subtract 50: 40 = 40 sin((4π/5) + C).Divide both sides by 40: 1 = sin((4π/5) + C).The sine function equals 1 at π/2 + 2πk, where k is an integer.So, (4π/5) + C = π/2 + 2πk.Solving for C:C = π/2 - 4π/5 + 2πk.Let's compute π/2 - 4π/5:Convert to common denominator, which is 10:π/2 = 5π/104π/5 = 8π/10So, 5π/10 - 8π/10 = -3π/10.Therefore, C = -3π/10 + 2πk.Since the sine function is periodic, we can choose k=0 for the principal value.Thus, C = -3π/10.Alternatively, we can express this as C = (7π/10) if we add 2π, but since it's a phase shift, both are correct, but let's stick with the negative value for simplicity.So, C = -3π/10.Let me verify this with the other point, the minimum at t=9.Compute Q(9):Q(9) = 40 sin((π/5)(9) - 3π/10) + 50.First, compute the argument:(9π/5) - (3π/10) = (18π/10 - 3π/10) = 15π/10 = 3π/2.sin(3π/2) = -1.So, Q(9) = 40*(-1) + 50 = -40 + 50 = 10. Correct.Similarly, let's check t=4:Argument: (4π/5) - 3π/10 = (8π/10 - 3π/10) = 5π/10 = π/2.sin(π/2) = 1.So, Q(4) = 40*1 + 50 = 90. Correct.Therefore, the function is Q(t) = 40 sin((π/5)t - 3π/10) + 50.So, the values are A=40, B=π/5, C=-3π/10, D=50.Let me just recap:1) For the quadratic sequence, we set up three equations based on the given participants and solved the system to get a=2, b=1, c=12.2) For the positivity function, we used the max and min to find A and D, then used the time between max and min to find the period and hence B. Then, using the max point, we solved for C.Everything seems consistent when checked with the given data points.Final Answer1. The quadratic function is ( P(n) = 2n^2 + n + 12 ), so the constants are ( a = boxed{2} ), ( b = boxed{1} ), and ( c = boxed{12} ).2. The positivity function is ( Q(t) = 40 sinleft(frac{pi}{5}t - frac{3pi}{10}right) + 50 ), so the constants are ( A = boxed{40} ), ( B = boxed{dfrac{pi}{5}} ), ( C = boxed{-dfrac{3pi}{10}} ), and ( D = boxed{50} ).</think>

🧠:<think>Alright, so I've got these two math problems here, and I need to figure them out step by step. Let me start with the first one.Problem 1: Permutations of Tone ChangesOkay, the first problem is about a visually impaired individual who listens to a radio show. The show is an hour long, and the host changes their tone every 15 minutes, which divides the show into four equal parts. The listener uses these tone changes to mentally segment the content into four parts, labeled A, B, C, and D. The question is asking how many different ways the listener can record the sequence of tone changes as a permutation of these letters, with each letter appearing exactly once.Hmm, so I think this is a permutations problem. Since there are four segments, each corresponding to a letter, and each letter must appear exactly once, we're looking at the number of ways to arrange four distinct items. I remember that the number of permutations of n distinct items is n factorial, which is n! So for four letters, it should be 4! = 4 × 3 × 2 × 1 = 24.Wait, let me make sure. Each tone change marks the start of a new segment, and the listener is noting the sequence. So if they start from the beginning, the first tone change is the end of segment A, the second is the end of segment B, and so on. But the problem says "the sequence of tone changes as a permutation of the letters A, B, C, and D." So each tone change corresponds to a segment, and the listener is noting the order in which the segments occur.But since the show is fixed, the segments are in order A, B, C, D. So does that mean the permutation is fixed? Wait, no, the problem says "how many different ways can the listener record the sequence of tone changes as a permutation of the letters A, B, C, and D." So maybe the listener is not necessarily starting at the beginning? Or perhaps the segments can be rearranged?Wait, no, the show is a series of educational programs, so each episode is fixed. So the segments are in order A, B, C, D. So the tone changes happen every 15 minutes, so the listener hears the tone changes in the order A, B, C, D. So the sequence is fixed, right? So why would the permutation vary?Wait, maybe I'm misunderstanding. Maybe the listener doesn't know the order of the segments, so they are trying to figure out the order based on the tone changes. But no, the problem says the listener uses the tone changes to mentally segment the content into four quarters. So they know that each tone change signifies the end of a segment, but perhaps they don't know the order of the segments? Or maybe the segments are not in order?Wait, the problem says "the listener starts listening to the show from the beginning." So they start at the beginning, and every 15 minutes, the host changes tone, which the listener uses to segment the content into four parts. So the first 15 minutes is segment A, the next 15 minutes is segment B, and so on. So the order is fixed as A, B, C, D.But the question is about how many different ways the listener can record the sequence of tone changes as a permutation of the letters A, B, C, and D. Hmm, that's confusing. If the segments are in order, then the sequence of tone changes is fixed as A, B, C, D. So why would there be different permutations?Wait, maybe the listener doesn't know the order of the segments, so they have to figure out the order based on the content. But the problem doesn't mention anything about the content being shuffled or anything. It just says the listener uses the tone changes to segment the content into four quarters. So each tone change signifies the end of a segment, but the segments themselves are in order.Wait, maybe the listener is trying to label each segment with a letter, but the letters can be assigned in any order? So for example, the first segment could be labeled A, B, C, or D, and so on for each subsequent segment. So the listener is assigning labels to each segment, and the question is how many different ways can they assign the labels such that each letter is used exactly once.Oh, that makes sense. So the listener is assigning the labels A, B, C, D to the four segments, but they don't know which segment corresponds to which label. So they have to figure out the permutation of the labels. But since they start listening from the beginning, they can assign the labels in any order, so the number of permutations is 4!.So that would be 24 different ways. So the answer is 24.Wait, let me just confirm. If the segments are in order, but the listener is assigning labels to each segment, and each label must be used exactly once, then yes, it's 4! = 24.Okay, that seems right.Problem 2: Percentage Reduction in Unique WordsAlright, moving on to the second problem. The listener creates mathematical patterns based on the frequency of specific words used during the show. The first segment (A) uses exactly 100 different unique words. In each subsequent segment (B, C, D), the number of unique words used is reduced by a constant percentage compared to the previous segment. By the end of the show, the total number of unique words used across all segments is 232. We need to find the percentage reduction from one segment to the next.So, let's break this down. The first segment has 100 unique words. Each subsequent segment has a number of unique words that is a constant percentage less than the previous one. So, this is a geometric sequence where each term is multiplied by a common ratio, which is (1 - r), where r is the percentage reduction.Wait, actually, if the number is reduced by a constant percentage, say p%, then the number of words in the next segment is previous * (1 - p/100). So, the ratio is (1 - p/100). Let's denote this ratio as r, so r = 1 - p/100.So, the number of unique words in each segment is:Segment A: 100Segment B: 100 * rSegment C: 100 * r^2Segment D: 100 * r^3The total number of unique words is the sum of these four segments: 100 + 100r + 100r^2 + 100r^3 = 232.So, we can write the equation:100(1 + r + r^2 + r^3) = 232Divide both sides by 100:1 + r + r^2 + r^3 = 2.32So, we need to solve for r in the equation:r^3 + r^2 + r + 1 = 2.32Which simplifies to:r^3 + r^2 + r - 1.32 = 0Hmm, solving a cubic equation. That might be a bit tricky. Maybe I can approximate it or find a rational root.Let me see if I can guess a value for r. Since the total is 232, which is more than 200, the reduction can't be too high. Let's try r = 0.9, which would mean a 10% reduction each time.Plugging r = 0.9:0.9^3 + 0.9^2 + 0.9 + 1 = 0.729 + 0.81 + 0.9 + 1 = 3.439, which is way higher than 2.32. So, 0.9 is too high.Wait, actually, if r is the ratio, and the total is 2.32, which is less than 4, so r must be less than 1, but how much less?Wait, let's try r = 0.8:0.8^3 + 0.8^2 + 0.8 + 1 = 0.512 + 0.64 + 0.8 + 1 = 2.952, still higher than 2.32.Hmm, need a lower r. Let's try r = 0.7:0.7^3 + 0.7^2 + 0.7 + 1 = 0.343 + 0.49 + 0.7 + 1 = 2.533, still higher.r = 0.6:0.6^3 + 0.6^2 + 0.6 + 1 = 0.216 + 0.36 + 0.6 + 1 = 2.176, which is less than 2.32.So, between 0.6 and 0.7.We have:At r = 0.6: sum = 2.176At r = 0.7: sum = 2.533We need sum = 2.32So, let's try r = 0.65:0.65^3 + 0.65^2 + 0.65 + 1Calculate each term:0.65^3 = 0.2746250.65^2 = 0.42250.65 = 0.651 = 1Sum: 0.274625 + 0.4225 + 0.65 + 1 = 2.347125That's pretty close to 2.32. So, 2.347125 is a bit higher than 2.32.So, let's try r = 0.64:0.64^3 = 0.2621440.64^2 = 0.40960.64 = 0.641 = 1Sum: 0.262144 + 0.4096 + 0.64 + 1 = 2.311744That's just below 2.32.So, at r = 0.64, sum ≈ 2.3117At r = 0.65, sum ≈ 2.3471We need sum = 2.32So, let's find r between 0.64 and 0.65.Let me set up a linear approximation.Let’s denote f(r) = r^3 + r^2 + r + 1We have f(0.64) ≈ 2.3117f(0.65) ≈ 2.3471We need f(r) = 2.32So, the difference between f(0.64) and f(0.65) is 2.3471 - 2.3117 = 0.0354We need to cover 2.32 - 2.3117 = 0.0083So, the fraction is 0.0083 / 0.0354 ≈ 0.234So, r ≈ 0.64 + 0.234*(0.65 - 0.64) = 0.64 + 0.00234 ≈ 0.64234So, approximately 0.6423Therefore, r ≈ 0.6423So, the ratio is approximately 0.6423, which means the reduction is 1 - 0.6423 = 0.3577, or 35.77%Wait, but let me check:If r = 0.6423, then:0.6423^3 + 0.6423^2 + 0.6423 + 1Calculate each term:0.6423^3 ≈ 0.6423 * 0.6423 * 0.6423First, 0.6423 * 0.6423 ≈ 0.4125Then, 0.4125 * 0.6423 ≈ 0.26460.6423^2 ≈ 0.41250.6423 ≈ 0.64231 = 1Sum: 0.2646 + 0.4125 + 0.6423 + 1 ≈ 2.3194That's pretty close to 2.32. So, r ≈ 0.6423Therefore, the percentage reduction is (1 - r) * 100 ≈ (1 - 0.6423) * 100 ≈ 35.77%So, approximately 35.77% reduction.But let me see if I can get a more accurate value.Alternatively, maybe I can set up the equation:r^3 + r^2 + r + 1 = 2.32So, r^3 + r^2 + r = 1.32Let me try r = 0.642:0.642^3 ≈ 0.642 * 0.642 * 0.6420.642 * 0.642 ≈ 0.4121640.412164 * 0.642 ≈ 0.26420.642^2 ≈ 0.4121640.642 ≈ 0.642Sum: 0.2642 + 0.412164 + 0.642 ≈ 1.318364Which is very close to 1.32. So, r ≈ 0.642So, the reduction is 1 - 0.642 = 0.358, or 35.8%So, approximately 35.8% reduction.But let me check with r = 0.642:Sum: 1 + 0.642 + 0.642^2 + 0.642^3 ≈ 1 + 0.642 + 0.412 + 0.264 ≈ 2.318, which is just under 2.32.So, maybe r is slightly higher than 0.642.Let me try r = 0.643:0.643^3 ≈ 0.643 * 0.643 * 0.6430.643 * 0.643 ≈ 0.4134490.413449 * 0.643 ≈ 0.26520.643^2 ≈ 0.4134490.643 ≈ 0.643Sum: 0.2652 + 0.413449 + 0.643 ≈ 1.3216So, 1 + 0.643 + 0.413449 + 0.2652 ≈ 2.3216Which is just over 2.32.So, the exact r is between 0.642 and 0.643.Since 2.318 at r=0.642 and 2.3216 at r=0.643, and we need 2.32.So, the difference between 2.3216 and 2.318 is 0.0036We need 2.32 - 2.318 = 0.002So, fraction is 0.002 / 0.0036 ≈ 0.555So, r ≈ 0.642 + 0.555*(0.643 - 0.642) ≈ 0.642 + 0.000555 ≈ 0.642555So, approximately 0.6426Therefore, r ≈ 0.6426So, percentage reduction is 1 - 0.6426 = 0.3574, or 35.74%So, approximately 35.74%But since the problem is asking for the percentage reduction, we can round it to two decimal places, so 35.74%, which is approximately 35.74%But maybe we can express it as a fraction or exact decimal.Alternatively, perhaps we can solve the cubic equation more accurately.But given that it's a percentage, and in the context of the problem, it's likely expecting a whole number or a simpler decimal.Wait, let me check if 35.74% is the correct answer.Wait, if we take r = 0.6426, then the total unique words would be:100*(1 + 0.6426 + 0.6426^2 + 0.6426^3) ≈ 100*(1 + 0.6426 + 0.413 + 0.265) ≈ 100*(2.3206) ≈ 232.06, which is very close to 232.So, that seems accurate.Therefore, the percentage reduction is approximately 35.74%, which we can round to 35.7% or 36%.But let me see if the problem expects an exact value or if it's okay with an approximate.Given that it's a percentage reduction, and the total is 232, which is a whole number, perhaps the exact value is a fraction that results in 232 when summed.Alternatively, maybe we can express it as a fraction.Wait, let's denote r as a fraction. Let me try to see if r is a simple fraction.Suppose r = 3/5 = 0.6, but we saw that gives a total of 2.176, which is too low.r = 4/5 = 0.8, which gives 2.952, too high.r = 5/8 = 0.625:0.625^3 + 0.625^2 + 0.625 + 1 = 0.244140625 + 0.390625 + 0.625 + 1 ≈ 2.26, which is still less than 2.32.r = 0.64:Sum ≈ 2.3117r = 0.65:Sum ≈ 2.3471So, it's between 0.64 and 0.65, which we've already calculated.Alternatively, maybe the problem expects us to use logarithms or another method, but given that it's a cubic equation, it's not straightforward.Alternatively, perhaps the problem is designed so that the ratio is 0.8, but that gives a higher total, so maybe not.Alternatively, maybe the problem is expecting us to use the formula for the sum of a geometric series.Wait, the sum S = a*(1 - r^n)/(1 - r), where a is the first term, r is the common ratio, and n is the number of terms.In this case, a = 100, n = 4, and S = 232.So, 232 = 100*(1 - r^4)/(1 - r)Divide both sides by 100:2.32 = (1 - r^4)/(1 - r)Multiply both sides by (1 - r):2.32*(1 - r) = 1 - r^4Expand left side:2.32 - 2.32r = 1 - r^4Bring all terms to one side:r^4 - 2.32r + 1.32 = 0Hmm, that's a quartic equation, which is even more complicated.Wait, but earlier I had:1 + r + r^2 + r^3 = 2.32Which is the same as:r^3 + r^2 + r - 1.32 = 0So, perhaps I can use the quartic equation approach, but it's more complex.Alternatively, maybe I can use the fact that r is close to 0.64, as we found earlier, and use that as an approximate solution.Given that, I think the percentage reduction is approximately 35.74%, which we can round to 35.7% or 36%.But let me check if 35.7% is acceptable or if it's better to present it as a fraction.Alternatively, maybe the problem expects an exact value, but given that it's a cubic equation, it's unlikely to have a simple exact solution.Therefore, I think the answer is approximately 35.7% reduction.But let me double-check my calculations.Wait, if r = 0.6426, then the total is approximately 232.06, which is very close to 232.So, the percentage reduction is 1 - 0.6426 = 0.3574, which is 35.74%.So, I think that's the answer.Final Answer1. boxed{24}2. boxed{35.7%}</think>