🧠:<think>Okay, so I have this problem where my colleague Alex uses white noise to cope with tinnitus. The effectiveness of the white noise is modeled by this function E(f, t) = A·sin(Bf)·e^(-Ct) + D. The constants are given as A=5, B=0.002, C=0.1, and D=10. Part 1 asks me to find the frequency f that maximizes E(f, t) when t is fixed at 30 minutes. The frequency range is from 100 Hz to 2000 Hz. Alright, so first, let me write down the function with the given constants. Plugging in A, B, C, D:E(f, t) = 5·sin(0.002f)·e^(-0.1t) + 10.Since t is fixed at 30 minutes, I can substitute t=30 into the equation:E(f, 30) = 5·sin(0.002f)·e^(-0.1*30) + 10.Calculating the exponential part first: e^(-0.1*30) = e^(-3). I know that e^(-3) is approximately 0.0498. So, the equation becomes:E(f, 30) ≈ 5·sin(0.002f)·0.0498 + 10.Multiplying 5 and 0.0498: 5*0.0498 ≈ 0.249. So,E(f, 30) ≈ 0.249·sin(0.002f) + 10.Now, to maximize E(f, 30), since 10 is a constant, I need to maximize the term 0.249·sin(0.002f). The maximum value of sin(x) is 1, so the maximum effectiveness would be 0.249*1 + 10 = 10.249.So, the maximum occurs when sin(0.002f) = 1. That happens when 0.002f = π/2 + 2πk, where k is an integer. Solving for f:0.002f = π/2 + 2πk f = (π/2 + 2πk)/0.002 f = (π/2)/0.002 + (2πk)/0.002 f = (π/2)/0.002 + 1000πk.Calculating (π/2)/0.002: π is approximately 3.1416, so π/2 ≈ 1.5708. Dividing by 0.002: 1.5708 / 0.002 = 785.4. So, f ≈ 785.4 + 1000πk.Now, since f must be between 100 Hz and 2000 Hz, let's find the k values that satisfy this.For k=0: f ≈ 785.4 Hz. That's within 100-2000 Hz.For k=1: f ≈ 785.4 + 1000*3.1416 ≈ 785.4 + 3141.6 ≈ 3927 Hz. That's above 2000 Hz, so too high.k=-1: f ≈ 785.4 - 3141.6 ≈ negative, which is below 100 Hz. So, only k=0 gives a valid frequency in the range.Therefore, the frequency that maximizes E(f, 30) is approximately 785.4 Hz.Wait, but let me double-check. The function is E(f, t) = 5·sin(0.002f)·e^(-0.1t) + 10. So, when t=30, the exponential term is e^(-3) ≈ 0.0498, as I had before. So, the function is 0.249·sin(0.002f) + 10.So, yes, to maximize this, sin(0.002f) should be 1. So, 0.002f = π/2 + 2πk. So, f = (π/2 + 2πk)/0.002.As above, for k=0, f≈785.4 Hz. For k=1, f≈785.4 + 1000π ≈785.4 + 3141.6≈3927 Hz, which is beyond 2000 Hz. So, only k=0 is valid.So, the optimal frequency is approximately 785.4 Hz. Since the problem asks for the frequency, I can write it as approximately 785 Hz, but maybe more precise.Alternatively, maybe I should express it more accurately. Let's compute 0.002f = π/2. So, f = (π/2)/0.002.Calculating π/2: approximately 1.57079632679.Divide by 0.002: 1.57079632679 / 0.002 = 785.398163395 Hz.So, approximately 785.4 Hz. So, 785.4 Hz is the exact value, but since frequencies are often given in whole numbers, maybe 785 Hz or 785.4 Hz. The problem doesn't specify, so I think 785.4 Hz is fine.So, that's part 1.Part 2: If Alex listens to the optimal frequency found in part 1, calculate the time t at which the effectiveness E(f, t) first drops to 50% of its initial value at t=0 minutes.So, first, let's find the initial effectiveness at t=0. Using the optimal frequency f=785.4 Hz.E(f, 0) = 5·sin(0.002*785.4)·e^(0) + 10.Compute sin(0.002*785.4): 0.002*785.4 = 1.5708, which is π/2. So, sin(π/2)=1. So,E(f, 0) = 5*1*1 + 10 = 5 + 10 = 15.So, the initial effectiveness is 15. 50% of that is 7.5.We need to find t such that E(f, t) = 7.5.So, set up the equation:5·sin(0.002f)·e^(-0.1t) + 10 = 7.5.But wait, f is fixed at 785.4 Hz, so sin(0.002f) is sin(π/2)=1. So, the equation simplifies to:5·1·e^(-0.1t) + 10 = 7.5.So,5e^(-0.1t) + 10 = 7.5.Subtract 10 from both sides:5e^(-0.1t) = -2.5.Wait, that can't be right. Because 5e^(-0.1t) is always positive, and adding 10, it's at least 10. So, 5e^(-0.1t) + 10 can't be less than 10. But 7.5 is less than 10. So, that suggests that the effectiveness can't drop below 10? Wait, but at t=0, E=15, and as t increases, e^(-0.1t) decreases, so E decreases. But since D=10, the effectiveness approaches 10 as t approaches infinity. So, the effectiveness can't drop below 10. So, 50% of 15 is 7.5, which is below 10, so it's impossible. Therefore, the effectiveness never drops to 50% of its initial value because it asymptotically approaches 10.Wait, that can't be. Maybe I made a mistake in interpreting the problem.Wait, the function is E(f, t) = 5·sin(Bf)·e^(-Ct) + D. So, when f is optimal, sin(Bf)=1, so E(f, t) = 5·1·e^(-0.1t) + 10.So, E(f, t) = 5e^(-0.1t) + 10.So, at t=0, E=15. As t increases, E decreases towards 10. So, 50% of 15 is 7.5, which is below 10. So, the effectiveness never reaches 7.5, because it only goes down to 10. Therefore, the time when E=7.5 is undefined, because it never happens.But that contradicts the problem statement, which says "calculate the time t at which the effectiveness E(f, t) first drops to 50% of its initial value at t=0 minutes."Wait, maybe I misread the function. Let me check again.E(f, t) = A·sin(Bf)·e^(-Ct) + D.A=5, B=0.002, C=0.1, D=10.So, when f is optimal, sin(Bf)=1, so E(f, t)=5·1·e^(-0.1t) + 10.So, E(t) = 5e^(-0.1t) + 10.So, initial value at t=0 is 15. 50% of 15 is 7.5. But E(t) approaches 10 as t approaches infinity. So, E(t) can't reach 7.5. Therefore, there is no such t where E(t)=7.5.But the problem says to calculate the time t when it first drops to 50% of its initial value. So, perhaps I made a mistake in the function.Wait, maybe the function is E(f, t) = A·sin(Bf + Ct) + D? No, the original problem says E(f, t) = A·sin(Bf)·e^(-Ct) + D.So, no, it's correct as I have it.Wait, perhaps the 50% is not of the initial value, but of the maximum effectiveness? But the problem says "50% of its initial value at t=0 minutes." So, initial value is 15, 50% is 7.5.But since E(t) approaches 10, it never reaches 7.5. Therefore, the time t does not exist. But the problem says to calculate it, so maybe I made a mistake.Wait, perhaps I misapplied the function. Let me double-check.E(f, t) = 5·sin(0.002f)·e^(-0.1t) + 10.At f=785.4 Hz, sin(0.002f)=1, so E(t)=5e^(-0.1t) + 10.So, E(t)=5e^(-0.1t) + 10.We need to find t such that E(t)=7.5.So,5e^(-0.1t) + 10 = 7.5 5e^(-0.1t) = -2.5.But 5e^(-0.1t) is always positive, so this equation has no solution. Therefore, the effectiveness never drops to 7.5.But the problem says to calculate the time when it first drops to 50% of its initial value. So, perhaps the problem is misinterpreted.Wait, maybe 50% of the maximum effectiveness, not the initial value. The maximum effectiveness is 15, so 50% is 7.5. But as above, it never reaches 7.5.Alternatively, maybe 50% of the difference between the maximum and the baseline. The baseline is D=10. The maximum is 15, so the difference is 5. 50% of that is 2.5, so 10 + 2.5=12.5. So, maybe the problem is asking when E(t)=12.5.But the problem says "50% of its initial value at t=0 minutes." So, initial value is 15, 50% is 7.5.Alternatively, maybe the problem is considering the amplitude part, which is 5·sin(Bf)·e^(-Ct). So, 5·sin(Bf)·e^(-Ct) is the varying part, and D=10 is the baseline. So, maybe 50% of the initial effectiveness is 50% of 5·sin(Bf), which is 2.5, so E(t)=10 + 2.5=12.5.But the problem says "50% of its initial value at t=0 minutes." So, initial value is 15, 50% is 7.5. So, perhaps the problem is incorrectly set, or I'm misinterpreting.Alternatively, maybe the function is E(f, t) = A·sin(Bf + Ct) + D, but no, the original problem says E(f, t) = A·sin(Bf)·e^(-Ct) + D.Wait, perhaps the problem is that the effectiveness is modeled as A·sin(Bf)·e^(-Ct) + D, so when f is optimal, sin(Bf)=1, so E(t)=A·e^(-Ct) + D. So, the effectiveness is A·e^(-Ct) + D.So, initial effectiveness is A + D = 5 + 10 = 15.We need to find t when E(t) = 0.5*15 = 7.5.So, 5e^(-0.1t) + 10 = 7.5 5e^(-0.1t) = -2.5.This is impossible because e^(-0.1t) is always positive, so 5e^(-0.1t) is positive, and adding 10, it's at least 10. So, E(t) can't be less than 10. Therefore, the effectiveness never drops to 7.5.So, perhaps the problem is asking for 50% of the maximum effectiveness, which is 15, so 7.5, but as we saw, it's impossible. Alternatively, maybe 50% of the effectiveness above the baseline. The baseline is 10, so 50% of 5 (since 15-10=5) is 2.5, so E(t)=10 + 2.5=12.5.So, let's try that. Let me assume that the problem meant 50% of the effectiveness above the baseline, so 12.5.So, set E(t)=12.5:5e^(-0.1t) + 10 = 12.5 5e^(-0.1t) = 2.5 e^(-0.1t) = 0.5 Take natural log of both sides:-0.1t = ln(0.5) t = ln(0.5)/(-0.1) ln(0.5) is approximately -0.6931, so:t = (-0.6931)/(-0.1) = 6.931 minutes.So, approximately 6.93 minutes.But the problem says "50% of its initial value at t=0 minutes." So, initial value is 15, 50% is 7.5. But as we saw, it's impossible. So, maybe the problem is misworded, or perhaps I'm misunderstanding.Alternatively, maybe the function is E(f, t) = A·sin(Bf)·e^(-Ct) + D, and when f is optimal, sin(Bf)=1, so E(t)=A·e^(-Ct) + D. So, the effectiveness is A·e^(-Ct) + D.So, the initial effectiveness is A + D = 15, and it decays to D=10 as t approaches infinity. So, the effectiveness can't drop below 10, so 50% of 15 is 7.5, which is below 10, so it's impossible.Therefore, perhaps the problem is asking for when the effectiveness drops to 50% of the maximum possible effectiveness, which is 15. So, 7.5 is 50% of 15, but as we saw, it's impossible.Alternatively, maybe the problem is asking for when the varying part (A·sin(Bf)·e^(-Ct)) drops to 50% of its initial value. The varying part at t=0 is A·sin(Bf)=5*1=5. So, 50% of 5 is 2.5. So, set A·sin(Bf)·e^(-Ct)=2.5:5·1·e^(-0.1t)=2.5 e^(-0.1t)=0.5 Same as before, t=6.931 minutes.So, perhaps that's what the problem is asking. So, the effectiveness function is E(t)=5e^(-0.1t)+10. The varying part is 5e^(-0.1t), which starts at 5 and decays. So, when it drops to 2.5, the total effectiveness is 10 + 2.5=12.5, which is 50% of the varying part's initial value.But the problem says "50% of its initial value at t=0 minutes." So, initial value is 15, so 50% is 7.5. But as we saw, it's impossible. Therefore, perhaps the problem is misworded, or I'm misinterpreting.Alternatively, maybe the problem is considering the effectiveness relative to the baseline. The baseline is 10, so the effectiveness is 15 at t=0, which is 5 above the baseline. So, 50% of that is 2.5, so the effectiveness is 10 + 2.5=12.5. So, that's the same as before, t≈6.93 minutes.Given that, I think the problem is asking for when the effectiveness drops to 12.5, which is 50% of the varying part above the baseline. So, t≈6.93 minutes.But to be precise, let's solve it step by step.Given E(t)=5e^(-0.1t)+10.We need to find t such that E(t)=7.5. But as we saw, this is impossible because E(t) approaches 10 from above.Therefore, perhaps the problem is asking for when the effectiveness drops to 50% of the maximum effectiveness, which is 15. So, 50% is 7.5, but it's impossible. Therefore, maybe the problem is asking for when the effectiveness drops to 50% of the initial effectiveness above the baseline. The initial effectiveness above the baseline is 5, so 50% is 2.5, so E(t)=10 + 2.5=12.5.So, solving 5e^(-0.1t) + 10 = 12.5:5e^(-0.1t) = 2.5 e^(-0.1t) = 0.5 Take natural log:-0.1t = ln(0.5) t = ln(0.5)/(-0.1) t = (-0.6931)/(-0.1) t ≈ 6.931 minutes.So, approximately 6.93 minutes.Therefore, the time t is approximately 6.93 minutes.But to confirm, let's plug t=6.931 into E(t):E(t)=5e^(-0.1*6.931)+10 =5e^(-0.6931)+10 =5*(0.5)+10 =2.5 + 10 =12.5.Yes, that's correct.So, even though the problem says "50% of its initial value at t=0 minutes," which would be 7.5, which is impossible, I think the intended interpretation is 50% of the effectiveness above the baseline, which is 12.5, achieved at t≈6.93 minutes.Therefore, the answer is approximately 6.93 minutes.But to be precise, let's calculate it more accurately.ln(0.5) is exactly -ln(2) ≈ -0.69314718056.So, t = (-ln(2))/(-0.1) = ln(2)/0.1 ≈ 0.69314718056/0.1 ≈ 6.9314718056 minutes.So, approximately 6.9315 minutes.Rounding to a reasonable decimal place, maybe 6.93 minutes or 6.9315 minutes.Alternatively, if we need to express it in minutes and seconds, 0.9315 minutes is approximately 55.89 seconds, so 6 minutes and 56 seconds. But the problem doesn't specify, so probably decimal minutes is fine.So, summarizing:1. The optimal frequency is approximately 785.4 Hz.2. The time when the effectiveness drops to 50% of its initial value (interpreted as 50% of the varying part above the baseline) is approximately 6.93 minutes.But wait, let me double-check the initial interpretation. The problem says "50% of its initial value at t=0 minutes." So, initial value is 15, 50% is 7.5. But as we saw, E(t) can't reach 7.5 because it approaches 10. So, perhaps the problem is incorrectly set, or I'm misinterpreting.Alternatively, maybe the function is E(f, t) = A·sin(Bf + Ct) + D, but no, the original problem says E(f, t) = A·sin(Bf)·e^(-Ct) + D.Wait, perhaps the problem is considering the effectiveness as a percentage of the maximum effectiveness, which is 15. So, 50% of 15 is 7.5, but as we saw, it's impossible. Therefore, perhaps the problem is asking for when the effectiveness drops to 50% of the maximum possible effectiveness, which is 15, but it's impossible because it can't go below 10.Alternatively, maybe the problem is considering the effectiveness relative to the baseline. The baseline is 10, so the effectiveness is 15 at t=0, which is 5 above the baseline. So, 50% of that is 2.5, so the effectiveness is 10 + 2.5=12.5. So, that's the same as before, t≈6.93 minutes.Given that, I think the problem is asking for when the effectiveness drops to 12.5, which is 50% of the varying part above the baseline. So, t≈6.93 minutes.Therefore, the answers are:1. f ≈ 785.4 Hz2. t ≈ 6.93 minutesBut to be precise, let's write the exact expressions.For part 1, f = (π/2)/0.002 = π/(2*0.002) = π/0.004 = 250π ≈ 785.398 Hz.For part 2, t = ln(0.5)/(-0.1) = ln(2)/0.1 ≈ 6.93147 minutes.So, exact expressions:1. f = 250π Hz ≈ 785.4 Hz2. t = (ln 2)/0.1 ≈ 6.931 minutesTherefore, the answers are:1. The optimal frequency is 250π Hz, approximately 785.4 Hz.2. The time is (ln 2)/0.1 minutes, approximately 6.931 minutes.So, I think that's the solution.</think>

🧠:<think>Okay, so I have this math problem about Charlotte, a playwright who used the Fibonacci sequence in her play. The play has 24 acts, and each act's number of lines is the nth Fibonacci number. I need to find two things: the total number of lines in the entire play and the prime factors of that total sum.First, let me recall what the Fibonacci sequence is. It starts with 0 and 1, and each subsequent number is the sum of the two preceding ones. So, the sequence goes 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on. But wait, in the problem, it says the nth Fibonacci number for each act. So, does that mean the first act is the 1st Fibonacci number, which is 0 or 1? Hmm, sometimes Fibonacci sequences are indexed starting at 0, so F0=0, F1=1, F2=1, etc. But sometimes people start at F1=1, F2=1, F3=2, etc. I need to clarify this.The problem says "the nth Fibonacci number, where n is the act number." So, if n=1, is it F1 or F0? Since the first act is n=1, I think it's safer to assume that the first Fibonacci number is 1, so F1=1, F2=1, F3=2, etc. Because if it started at 0, the first act would have 0 lines, which doesn't make much sense for a play. So, I'll proceed with F1=1, F2=1, F3=2, and so on.So, for each act from 1 to 24, the number of lines is the nth Fibonacci number. Therefore, I need to compute the sum of the first 24 Fibonacci numbers, starting from F1=1.I remember that there's a formula for the sum of the first n Fibonacci numbers. Let me recall it. I think it's something like the (n+2)th Fibonacci number minus 1. Let me check that.If I consider the sum S = F1 + F2 + F3 + ... + Fn. Then, S = F(n+2) - 1. Let me test this with small n.For n=1: S=1. F(3)=2. 2-1=1. Correct.For n=2: S=1+1=2. F(4)=3. 3-1=2. Correct.For n=3: S=1+1+2=4. F(5)=5. 5-1=4. Correct.Okay, so the formula seems to hold. Therefore, the sum of the first 24 Fibonacci numbers is F(26) - 1.So, I need to find F26. Let me list out the Fibonacci numbers up to F26.Starting from F1=1:F1 = 1F2 = 1F3 = 2F4 = 3F5 = 5F6 = 8F7 = 13F8 = 21F9 = 34F10 = 55F11 = 89F12 = 144F13 = 233F14 = 377F15 = 610F16 = 987F17 = 1597F18 = 2584F19 = 4181F20 = 6765F21 = 10946F22 = 17711F23 = 28657F24 = 46368F25 = 75025F26 = 121393So, F26 is 121,393. Therefore, the sum S = 121,393 - 1 = 121,392.So, the total number of lines in the entire play is 121,392.Now, moving on to the second part: Charlotte used the prime factors of this sum as a cipher key. So, I need to find the prime factors of 121,392.First, let's factorize 121,392.I can start by dividing by small primes.121,392 is even, so divide by 2:121,392 ÷ 2 = 60,696Still even, divide by 2 again:60,696 ÷ 2 = 30,348Still even, divide by 2:30,348 ÷ 2 = 15,174Still even, divide by 2:15,174 ÷ 2 = 7,587Now, 7,587 is odd. Let's check divisibility by 3. Sum of digits: 7+5+8+7=27, which is divisible by 3.7,587 ÷ 3 = 2,5292,529: sum of digits 2+5+2+9=18, divisible by 3.2,529 ÷ 3 = 843843: 8+4+3=15, divisible by 3.843 ÷ 3 = 281Now, 281 is a prime number? Let me check.Check divisibility by primes up to sqrt(281) ≈ 16.76.281 ÷ 2 ≠ integer281 ÷ 3: 3*93=279, remainder 2281 ÷ 5: ends with 1, no281 ÷ 7: 7*40=280, remainder 1281 ÷ 11: 11*25=275, remainder 6281 ÷ 13: 13*21=273, remainder 8281 ÷ 17: 17*16=272, remainder 9So, 281 is prime.So, putting it all together, the prime factors are:2^4 * 3^3 * 281^1So, the prime factors are 2, 3, and 281.Wait, let me double-check the factorization steps.Starting with 121,392:121,392 ÷ 2 = 60,69660,696 ÷ 2 = 30,34830,348 ÷ 2 = 15,17415,174 ÷ 2 = 7,5877,587 ÷ 3 = 2,5292,529 ÷ 3 = 843843 ÷ 3 = 281281 is prime.Yes, that seems correct.So, the prime factors are 2, 3, and 281.Therefore, the key to the cipher is these prime factors.Final Answer1. The total number of lines is boxed{121392}.2. The prime factors are boxed{2}, boxed{3}, and boxed{281}.</think>

🧠:<think>First, I need to calculate the expected annual return of the original portfolio. The portfolio is divided into bonds and stocks, with 60% in bonds and 40% in stocks. The total value of the portfolio is 500,000.For the bonds:- 60% of 500,000 is 300,000.- The annual return on bonds is 3%, so the return from bonds is 300,000 multiplied by 0.03, which equals 9,000.For the stocks:- 40% of 500,000 is 200,000.- The annual return on stocks is 8%, so the return from stocks is 200,000 multiplied by 0.08, which equals 16,000.Adding the returns from bonds and stocks gives the total expected annual return of 25,000.Next, the relative decides to reallocate 10% of the portfolio from stocks to bonds. 10% of 500,000 is 50,000. This means the new allocation will be:- Bonds: 60% + 10% = 70% of 500,000, which is 350,000.- Stocks: 40% - 10% = 30% of 500,000, which is 150,000.Calculating the new returns:- Bonds: 350,000 multiplied by 0.03 equals 10,500.- Stocks: 150,000 multiplied by 0.08 equals 12,000.Adding these new returns gives the updated expected annual return of 22,500.</think>

🧠:<think>Okay, so I need to solve these two problems about a comic book editor planning the release schedule. Let me take them one at a time.Starting with the first problem: The editor wants to find the optimal interval ( t ) between releases that maximizes the average reader engagement ( E(t) ). The function given is ( E(t) = k e^{-lambda t} ), with ( k = 10 ) and ( lambda = 0.5 ). Hmm, so I need to maximize this function with respect to ( t ).Wait, but ( E(t) ) is an exponential decay function, right? Because as ( t ) increases, ( e^{-lambda t} ) decreases. So, the engagement decreases as the time between releases increases. Conversely, as ( t ) decreases, engagement increases. So, does that mean the maximum engagement is achieved when ( t ) is as small as possible?But hold on, the problem says "the interval between releases decreases," so ( t ) is the time between consecutive releases. If ( t ) is smaller, releases are more frequent. So, engagement increases as ( t ) decreases. But is there a lower bound on ( t )? The problem doesn't specify any constraints on ( t ), so theoretically, ( t ) could approach zero, making ( E(t) ) approach ( k ), which is 10. But in reality, there must be some practical constraints, like production time or something, but since the first problem doesn't mention that, maybe I just need to find the mathematical maximum.Wait, but if ( E(t) = 10 e^{-0.5 t} ), then the derivative of ( E(t) ) with respect to ( t ) is ( E'(t) = -5 e^{-0.5 t} ). Setting the derivative equal to zero to find maxima or minima: ( -5 e^{-0.5 t} = 0 ). But ( e^{-0.5 t} ) is always positive, so this equation has no solution. That means the function doesn't have a maximum in the domain of positive ( t ); it's always decreasing. Therefore, the maximum engagement occurs at the smallest possible ( t ).But without any constraints, the optimal ( t ) would be approaching zero. But that doesn't make sense in a real-world scenario because you can't release a comic book every fraction of a week. Maybe the problem expects me to realize that the function is always decreasing, so the maximum engagement is at the smallest feasible ( t ). But since the problem doesn't specify any constraints, perhaps it's a trick question where the maximum is at ( t = 0 ), but that would mean releasing all comics at once, which isn't practical.Wait, maybe I misinterpreted the function. Let me read it again: "the average reader engagement for a sneak peek increases exponentially as the interval between releases decreases." So, as ( t ) decreases, engagement increases. So, the function should be increasing as ( t ) decreases, which is consistent with ( E(t) = k e^{-lambda t} ) because as ( t ) decreases, ( -lambda t ) becomes less negative, so ( e^{-lambda t} ) increases. So, yes, the function is increasing as ( t ) decreases.Therefore, to maximize ( E(t) ), we need to minimize ( t ). But without any lower bound, the maximum is at ( t ) approaching zero. However, in the second problem, there are constraints on the workload, so maybe the first problem is just theoretical, and we can say that the optimal ( t ) is as small as possible, but perhaps the problem expects a specific value.Wait, maybe I'm supposed to find the maximum of the function ( E(t) ), but since it's an exponential decay, it doesn't have a maximum except at ( t = 0 ). So, perhaps the answer is that the engagement is maximized when ( t ) is as small as possible, but since the problem doesn't specify constraints, maybe we can't determine a specific numerical value. But the problem says "find the optimal interval ( t )", so maybe I'm missing something.Wait, perhaps I misread the function. Let me check again: ( E(t) = k e^{-lambda t} ). So, yes, it's an exponential decay. So, the maximum is at ( t = 0 ). But in reality, ( t ) can't be zero because you need some time to produce the comic. So, maybe the problem is expecting me to realize that the function doesn't have a maximum for ( t > 0 ), but rather, it's always increasing as ( t ) decreases. Therefore, the optimal ( t ) is the smallest possible feasible value, but since no constraints are given, perhaps the answer is that the engagement is maximized as ( t ) approaches zero.But that seems a bit odd. Maybe I should consider that the function might have been intended to be increasing with ( t ), but it's written as decreasing. Alternatively, perhaps the function is ( E(t) = k e^{lambda t} ), which would increase as ( t ) increases, but that contradicts the problem statement. Wait, the problem says "increases exponentially as the interval between releases decreases," so as ( t ) decreases, engagement increases. So, the function should be increasing when ( t ) decreases, which is consistent with ( E(t) = k e^{-lambda t} ) because as ( t ) decreases, ( E(t) ) increases.So, yes, the function is correct. Therefore, the maximum engagement is achieved as ( t ) approaches zero. But since the problem asks for the optimal interval ( t ), maybe it's expecting me to say that the optimal ( t ) is the smallest possible, but without constraints, it's zero. But that's not practical. Maybe the problem expects me to find the maximum of the function, but since it's always increasing as ( t ) decreases, the maximum is at the minimal ( t ). But without knowing the minimal ( t ), perhaps the answer is that the optimal ( t ) is as small as possible, but since no constraints are given, we can't determine a specific value. Hmm, this is confusing.Wait, maybe I'm overcomplicating it. Let me think again. The function ( E(t) = 10 e^{-0.5 t} ) is a decreasing function of ( t ). So, to maximize ( E(t) ), we need to minimize ( t ). Therefore, the optimal ( t ) is the smallest possible value. But since the problem doesn't specify any constraints, perhaps the answer is that the optimal interval is as small as possible, but without a specific lower bound, we can't give a numerical answer. However, maybe the problem expects me to realize that the function doesn't have a maximum in the domain ( t > 0 ), so the maximum is at ( t = 0 ), but that's not practical. Alternatively, perhaps I made a mistake in interpreting the function.Wait, maybe the function is supposed to be increasing with ( t ), but the problem says it increases as the interval decreases. So, the function is correct. Therefore, the maximum is at ( t = 0 ), but that's not feasible. So, perhaps the answer is that the optimal interval is as small as possible, but without constraints, we can't determine a specific value. But the problem says "find the optimal interval ( t )", so maybe I'm supposed to say that the engagement is maximized when ( t ) is minimized, but since no constraints are given, the optimal ( t ) is approaching zero.But maybe I'm missing something. Let me check the problem again: "the average reader engagement for a sneak peek increases exponentially as the interval between releases decreases." So, as ( t ) decreases, engagement increases. Therefore, the function ( E(t) = 10 e^{-0.5 t} ) is correct because as ( t ) decreases, ( E(t) ) increases. So, the maximum engagement is achieved when ( t ) is as small as possible. But without any constraints, the optimal ( t ) is zero, but that's not practical. So, perhaps the answer is that the optimal interval is as small as possible, but since no constraints are given, we can't determine a specific value. However, the problem might expect me to realize that the function doesn't have a maximum for ( t > 0 ), so the optimal ( t ) is the smallest feasible value, but without knowing the feasible range, we can't specify it.Wait, but the second problem does give constraints, so maybe the first problem is just theoretical, and the answer is that the optimal ( t ) is approaching zero. But that seems odd. Alternatively, maybe I'm supposed to find the maximum of the function, but since it's always increasing as ( t ) decreases, the maximum is at ( t = 0 ). So, perhaps the answer is ( t = 0 ), but that's not practical. Maybe the problem expects me to say that the optimal interval is as small as possible, but without constraints, we can't determine a specific value.Wait, perhaps I'm overcomplicating it. Maybe the function is supposed to have a maximum at some positive ( t ). Let me check the derivative again. ( E(t) = 10 e^{-0.5 t} ), so ( E'(t) = -5 e^{-0.5 t} ). Setting ( E'(t) = 0 ) gives no solution because ( e^{-0.5 t} ) is always positive. Therefore, the function is always decreasing, so the maximum is at ( t = 0 ). Therefore, the optimal interval is ( t = 0 ), but that's not practical. So, perhaps the answer is that the optimal interval is as small as possible, but without constraints, we can't determine a specific value.Wait, but the problem says "find the optimal interval ( t )", so maybe I'm supposed to say that the optimal interval is the smallest possible, but without constraints, it's zero. But that doesn't make sense in reality. Maybe the problem is expecting me to realize that the function doesn't have a maximum for ( t > 0 ), so the optimal ( t ) is approaching zero. Therefore, the answer is that the optimal interval is as small as possible, but since no constraints are given, we can't specify a numerical value. However, the problem might expect me to say that the optimal ( t ) is zero, but that's not practical.Wait, maybe I'm misinterpreting the function. Let me think again. If the engagement increases as the interval decreases, then the function should be increasing as ( t ) decreases, which is what ( E(t) = 10 e^{-0.5 t} ) does. So, the function is correct. Therefore, the maximum engagement is achieved when ( t ) is as small as possible. So, the optimal interval is the smallest possible ( t ). But without any constraints, we can't determine a specific value. Therefore, the answer is that the optimal interval is as small as possible, but since no constraints are given, we can't specify it numerically.Wait, but the problem gives ( k = 10 ) and ( lambda = 0.5 ), so maybe I'm supposed to find the maximum value of ( E(t) ), which is 10, achieved as ( t ) approaches zero. So, the optimal interval is approaching zero weeks, but that's not practical. Therefore, perhaps the answer is that the optimal interval is as small as possible, but without constraints, we can't determine a specific value.Wait, maybe the problem is expecting me to realize that the function is always increasing as ( t ) decreases, so the optimal ( t ) is the smallest feasible value, but since no constraints are given, the answer is that the optimal interval is approaching zero weeks. So, I think that's the answer.Now, moving on to the second problem: The production team can handle a maximum of ( P ) pages per week. Each comic book consists of ( N ) pages, and the team needs ( M ) additional weeks to complete the behind-the-scenes insights for each book. The total workload ( W ) in pages per week is given by ( W = frac{N}{t} + M ). The production team can handle up to 30 pages per week. Given ( N = 60 ) pages and ( M = 2 ) weeks, find the range of ( t ) that ensures the production team does not exceed their maximum workload.So, we have ( W = frac{60}{t} + 2 leq 30 ). We need to solve for ( t ).Let me write that inequality:( frac{60}{t} + 2 leq 30 )Subtract 2 from both sides:( frac{60}{t} leq 28 )Multiply both sides by ( t ) (assuming ( t > 0 )):( 60 leq 28 t )Divide both sides by 28:( t geq frac{60}{28} )Simplify the fraction:( t geq frac{15}{7} ) weeks, which is approximately 2.1429 weeks.So, the feasible range of ( t ) is ( t geq frac{15}{7} ) weeks.But wait, is there an upper limit on ( t )? The problem doesn't specify any maximum interval, so theoretically, ( t ) can be as large as possible, but in reality, there might be other constraints like reader engagement, but since the second problem only asks about the production workload, the upper limit is not specified. Therefore, the feasible range is ( t geq frac{15}{7} ) weeks.But let me double-check my steps:1. Start with ( W = frac{60}{t} + 2 leq 30 )2. Subtract 2: ( frac{60}{t} leq 28 )3. Multiply both sides by ( t ): ( 60 leq 28 t )4. Divide by 28: ( t geq frac{60}{28} = frac{15}{7} approx 2.1429 ) weeks.Yes, that seems correct. So, the feasible range of ( t ) is ( t geq frac{15}{7} ) weeks.But wait, let me think about the units. ( M = 2 ) weeks, but in the workload equation, it's added to ( frac{N}{t} ), which is pages per week. Wait, that doesn't make sense. Pages per week plus weeks? That would be adding apples and oranges. So, there must be a mistake in the problem statement or my interpretation.Wait, the problem says: "the total workload ( W ) in pages per week is given by ( W = frac{N}{t} + M )". But ( frac{N}{t} ) is pages per week, and ( M ) is weeks. So, adding them together would be adding pages per week and weeks, which is not dimensionally consistent. That must be a mistake.Wait, maybe I misread the problem. Let me check again: "the total workload ( W ) in pages per week is given by ( W = frac{N}{t} + M )". Hmm, that seems incorrect because ( M ) is in weeks, not pages per week. So, perhaps the problem meant that the workload is the sum of the pages per week from the comic books and the additional workload from the behind-the-scenes insights, but the units don't match.Alternatively, maybe ( M ) is the number of pages for the behind-the-scenes insights, but the problem says ( M ) is additional weeks. Wait, the problem says: "the team needs ( M ) additional weeks to complete the behind-the-scenes insights for each book." So, ( M ) is in weeks, not pages. Therefore, the equation ( W = frac{N}{t} + M ) is adding pages per week and weeks, which is not correct.This must be a mistake in the problem statement. Alternatively, perhaps ( M ) is the number of pages for the behind-the-scenes insights, but the problem says it's weeks. Hmm, this is confusing.Wait, maybe the workload ( W ) is the total number of pages per week, and ( M ) is the number of weeks required for the behind-the-scenes, so perhaps the total workload is the pages from the comic plus the pages from the behind-the-scenes divided by the time. But the problem says ( W = frac{N}{t} + M ), which is pages per week plus weeks, which doesn't make sense.Alternatively, maybe ( M ) is the number of pages for the behind-the-scenes, and the total workload is ( frac{N + M}{t} ). But the problem says ( W = frac{N}{t} + M ). Hmm.Wait, perhaps the problem meant that the workload is the pages per week from the comic plus the pages per week from the behind-the-scenes. But if ( M ) is weeks, then perhaps the behind-the-scenes workload is ( frac{M}{t} ) pages per week? That might make sense. So, maybe the equation should be ( W = frac{N}{t} + frac{M}{t} ), but the problem says ( W = frac{N}{t} + M ).Alternatively, maybe ( M ) is the number of pages for the behind-the-scenes, and the total workload is ( frac{N + M}{t} ). But the problem says ( W = frac{N}{t} + M ). Hmm.Wait, maybe the problem is correct, and ( M ) is the number of pages per week for the behind-the-scenes. But the problem says "the team needs ( M ) additional weeks to complete the behind-the-scenes insights for each book." So, ( M ) is weeks, not pages per week. Therefore, the equation ( W = frac{N}{t} + M ) is incorrect because it's adding pages per week and weeks.This is a problem. Maybe the correct equation should be ( W = frac{N}{t} + frac{M}{t} ), which would be pages per week. But the problem says ( W = frac{N}{t} + M ). Hmm.Alternatively, perhaps ( M ) is the number of pages for the behind-the-scenes, and the workload is ( frac{N + M}{t} ). But the problem says ( W = frac{N}{t} + M ).Wait, maybe the problem is correct, and ( M ) is the number of pages per week for the behind-the-scenes. But the problem says "the team needs ( M ) additional weeks to complete the behind-the-scenes insights for each book." So, ( M ) is weeks, not pages per week. Therefore, the equation is incorrect.This is confusing. Maybe I should proceed with the given equation, even though the units don't match, because that's what the problem states.So, given ( W = frac{60}{t} + 2 leq 30 ), even though the units are inconsistent, I'll proceed.So, solving ( frac{60}{t} + 2 leq 30 ):Subtract 2: ( frac{60}{t} leq 28 )Multiply both sides by ( t ): ( 60 leq 28 t )Divide by 28: ( t geq frac{60}{28} = frac{15}{7} approx 2.1429 ) weeks.So, the feasible range of ( t ) is ( t geq frac{15}{7} ) weeks.But I'm still concerned about the units. Maybe the problem intended ( M ) to be pages per week, but the problem says it's weeks. Alternatively, perhaps the equation should be ( W = frac{N + M}{t} ), but that's not what's given.Alternatively, maybe ( M ) is the number of pages for the behind-the-scenes, and the total workload is ( frac{N}{t} + frac{M}{t} ), but the problem says ( W = frac{N}{t} + M ).Hmm, maybe I should proceed with the given equation, even though it's dimensionally inconsistent, because that's what the problem states.Therefore, the feasible range of ( t ) is ( t geq frac{15}{7} ) weeks, approximately 2.14 weeks.So, summarizing:1. The optimal interval ( t ) that maximizes engagement is as small as possible, approaching zero weeks, but since no constraints are given, we can't specify a numerical value. However, if we consider the function mathematically, the maximum is at ( t = 0 ).2. The feasible range of ( t ) is ( t geq frac{15}{7} ) weeks, approximately 2.14 weeks.But wait, for the first problem, since the function is always increasing as ( t ) decreases, the optimal ( t ) is the smallest possible. But in the second problem, we have a lower bound on ( t ) to not exceed the workload. So, combining both, the optimal ( t ) that maximizes engagement while not exceeding workload is the smallest ( t ) allowed by the workload constraint, which is ( t = frac{15}{7} ) weeks.Wait, but the first problem is separate from the second. The first problem is just about maximizing engagement, regardless of workload, while the second is about ensuring the workload doesn't exceed the team's capacity. So, the first problem's answer is that the optimal ( t ) is as small as possible, but the second problem gives a constraint that ( t ) must be at least ( frac{15}{7} ) weeks.Therefore, the answers are:1. The optimal interval ( t ) is as small as possible, approaching zero weeks.2. The feasible range of ( t ) is ( t geq frac{15}{7} ) weeks.But perhaps the first problem is expecting a numerical answer, but since the function doesn't have a maximum for ( t > 0 ), the answer is that the optimal ( t ) is zero, but that's not practical. Alternatively, maybe the problem expects me to realize that the function is always increasing as ( t ) decreases, so the optimal ( t ) is the smallest feasible value, which would be determined by the second problem's constraint. But since the first problem is separate, I think the answer is that the optimal ( t ) is approaching zero weeks.But maybe I'm overcomplicating it. Let me check the first problem again: "find the optimal interval ( t ) that maximizes the average reader engagement ( E(t) )." Since ( E(t) ) is always increasing as ( t ) decreases, the optimal ( t ) is the smallest possible. But without constraints, it's zero. So, the answer is ( t = 0 ), but that's not practical. Therefore, perhaps the answer is that the optimal interval is as small as possible, but without constraints, we can't determine a specific value.Wait, but the problem gives specific values for ( k ) and ( lambda ), so maybe I'm supposed to find the maximum value of ( E(t) ), which is 10, achieved as ( t ) approaches zero. Therefore, the optimal interval is approaching zero weeks.So, to sum up:1. The optimal interval ( t ) is approaching zero weeks.2. The feasible range of ( t ) is ( t geq frac{15}{7} ) weeks.But I'm still unsure about the first problem because it's not practical to have ( t = 0 ). Maybe the problem expects me to realize that the function doesn't have a maximum for ( t > 0 ), so the optimal ( t ) is the smallest feasible value, which would be determined by other constraints, but since the first problem doesn't mention any, the answer is that the optimal ( t ) is as small as possible.Alternatively, maybe I'm supposed to find the maximum of the function, but since it's always increasing as ( t ) decreases, the maximum is at ( t = 0 ). So, the answer is ( t = 0 ).But in reality, ( t ) can't be zero, so perhaps the problem expects me to say that the optimal interval is as small as possible, but without constraints, we can't determine a specific value. However, since the problem gives specific values, maybe I'm supposed to find the maximum value of ( E(t) ), which is 10, achieved as ( t ) approaches zero.Therefore, the answer to the first problem is that the optimal interval ( t ) is approaching zero weeks, and the answer to the second problem is that ( t ) must be at least ( frac{15}{7} ) weeks.But wait, the second problem's equation is dimensionally inconsistent, which is confusing. Maybe I should proceed with the given equation despite the units issue.So, final answers:1. The optimal interval ( t ) is approaching zero weeks.2. The feasible range of ( t ) is ( t geq frac{15}{7} ) weeks, which is approximately 2.14 weeks.But since the first problem is separate, I think that's the answer.</think>