🧠:<think>Alright, so Professor Elara has this 1,000,000 to invest, and she's looking at two options: Steady Bonds and Practical Ventures Fund. She wants to maximize her expected return but also keep the risk manageable. The risk here is defined as the probability that the Practical Ventures Fund gives less than 3.5% return over five years, and she doesn't want that probability to exceed 10%. First, I need to tackle the first sub-problem: figuring out the maximum amount she can invest in the Practical Ventures Fund without exceeding that 10% probability threshold. Okay, so the Practical Ventures Fund has returns modeled by a normal distribution with a mean of 5% and a standard deviation of 2%. But wait, that's the annual return, right? Since she's investing for five years, I think I need to adjust the mean and standard deviation for the five-year period. For a normal distribution, when you compound returns over multiple periods, the mean return scales linearly with time, and the variance scales linearly as well. So, the mean return over five years would be 5% * 5 = 25%. The standard deviation would be 2% * sqrt(5). Let me calculate that: sqrt(5) is approximately 2.236, so 2% * 2.236 is about 4.472%. So, the five-year return for the Practical Ventures Fund is normally distributed with a mean of 25% and a standard deviation of approximately 4.472%. Now, she wants the probability that this return is less than 3.5% to be no more than 10%. So, we need to find the value of x such that P(R < 3.5%) ≤ 10%. Wait, actually, hold on. The 3.5% is the return from the Steady Bonds, which is fixed. So, she's comparing the Practical Ventures Fund's return to the Steady Bonds' return. But since she's investing in both, maybe I need to think about the total return. Hmm, no, the problem says the probability that the Practical Ventures Fund's return is less than 3.5% over five years. So, it's just about the Practical Ventures Fund's return, not the total return. So, we can model the Practical Ventures Fund's five-year return as N(25%, 4.472%). We need to find the probability that R < 3.5%, and set that probability to be ≤10%. To find this probability, we can standardize the normal variable. Let's denote Z as the standard normal variable. So, Z = (R - μ) / σ = (3.5 - 25) / 4.472 ≈ (-21.5) / 4.472 ≈ -4.806. Wait, that's a Z-score of approximately -4.806. Looking at standard normal distribution tables, the probability that Z is less than -4.806 is extremely low, way below 10%. In fact, it's practically zero. But that doesn't make sense because if she invests any amount in the Practical Ventures Fund, the probability of it underperforming the Steady Bonds is almost zero, which is way below 10%. So, does that mean she can invest the entire 1,000,000 in the Practical Ventures Fund? Wait, maybe I misunderstood the problem. Let me read it again. "the probability of the Practical Ventures Fund yielding a return less than 3.5% over the 5-year period being no more than 10%." So, she wants P(R < 3.5%) ≤ 10%. But as I calculated, with the five-year mean at 25% and standard deviation at ~4.472%, the Z-score for 3.5% is way in the left tail, giving a probability near zero. So, the probability is already way below 10%, regardless of how much she invests. Wait, but that can't be right because if she invests more in the Practical Ventures Fund, the total return is more variable, but the probability of the fund's return being less than 3.5% is independent of the amount invested. Hmm, actually, no. Wait, the return rate is independent of the amount invested. So, whether she invests 1 or 1,000,000, the probability that the return rate is less than 3.5% is the same. Therefore, the probability is fixed based on the fund's characteristics, not on how much she invests. So, if the probability is already less than 10%, she can invest the entire amount in the Practical Ventures Fund. But that seems contradictory because the problem is asking for the maximum amount x she can invest in the Practical Ventures Fund while keeping the probability ≤10%. Wait, maybe I made a mistake in calculating the five-year return. Let me double-check. The annual return is 5% with a standard deviation of 2%. For five years, the mean return is 5% * 5 = 25%. The standard deviation is 2% * sqrt(5) ≈ 4.472%. So, that part seems correct. So, the Z-score for 3.5% is (3.5 - 25)/4.472 ≈ -4.806. The probability of Z < -4.806 is effectively zero. So, the probability is already way below 10%, so she can invest the entire amount in the Practical Ventures Fund. But that seems too straightforward. Maybe the problem is considering the total return, not just the Practical Ventures Fund's return. Let me read the problem again. "the probability of the Practical Ventures Fund yielding a return less than 3.5% over the 5-year period being no more than 10%." So, it's specifically about the Practical Ventures Fund's return, not the total return. So, regardless of how much she invests, the probability that the fund's return is less than 3.5% is the same. Therefore, since that probability is already less than 10%, she can invest the entire amount. But wait, maybe the problem is considering the total return from both investments. Let me think. If she invests x in Practical Ventures and (1,000,000 - x) in Steady Bonds, the total return would be: Return from Steady Bonds: (1,000,000 - x) * (1 + 0.035)^5 Return from Practical Ventures: x * (1 + R), where R is normally distributed with mean 25% and standard deviation 4.472%. But the problem specifies the probability that the Practical Ventures Fund's return is less than 3.5%. So, it's not about the total return, but specifically the return from the Practical Ventures Fund. Therefore, the probability is solely based on the fund's characteristics, not on the amount invested. So, since the probability is already less than 10%, she can invest the entire amount. But that seems counterintuitive because usually, more investment in a volatile fund increases risk. But in this case, the risk is defined as the probability of the fund's return being less than 3.5%, which is independent of the investment amount. Wait, maybe I'm misunderstanding the definition of risk. Perhaps it's the probability that the total return is less than the return from investing all in Steady Bonds. Let me check the problem statement again. "the probability of the Practical Ventures Fund yielding a return less than 3.5% over the 5-year period being no more than 10%." No, it's specifically about the Practical Ventures Fund's return, not the total return. So, the amount invested doesn't affect the probability because the return rate is independent of the investment size. Therefore, the maximum x is 1,000,000. But that seems too straightforward, and the problem is asking for a sub-problem, so maybe I'm missing something. Wait, perhaps the return is in dollars, not in percentage. Let me re-examine the problem. The Practical Ventures Fund has a return modeled by R, which is a continuous random variable following a normal distribution with mean 5% and standard deviation 2%. So, R is in percentage terms. So, the return from the Practical Ventures Fund is R%, which is normally distributed with mean 5% and standard deviation 2% annually. Over five years, the mean return is 25%, standard deviation is 4.472%. So, the probability that R < 3.5% is the same regardless of the amount invested. Therefore, since this probability is already less than 10%, she can invest the entire amount. But let me verify the Z-score calculation. Z = (3.5 - 25)/4.472 ≈ (-21.5)/4.472 ≈ -4.806. Looking up Z = -4.806 in standard normal tables, the probability is approximately 0.000004, which is 0.0004%, way below 10%. Therefore, the probability is already way below 10%, so she can invest the entire 1,000,000 in the Practical Ventures Fund. But that seems too easy, so maybe I'm misinterpreting the problem. Perhaps the 3.5% is the total return, not the return from the Practical Ventures Fund. Let me read the problem again. "the probability of the Practical Ventures Fund yielding a return less than 3.5% over the 5-year period being no more than 10%." No, it's specifically about the Practical Ventures Fund's return. So, I think my initial conclusion is correct. Therefore, for part 1, the maximum x is 1,000,000. For part 2, calculating the total expected return. The expected return from the Practical Ventures Fund is 25%, so the expected amount from that is 1,000,000 * 1.25 = 1,250,000. The Steady Bonds have a fixed return of 3.5% annually, compounded annually. So, over five years, the amount is 1,000,000 * (1 + 0.035)^5. Calculating that: (1.035)^5 ≈ 1.187686So, the amount from Steady Bonds is approximately 1,187,686. But wait, if she invests the entire amount in Practical Ventures, she doesn't invest anything in Steady Bonds. So, the total expected return would be 1,250,000. But wait, if she invests x in Practical Ventures and (1,000,000 - x) in Steady Bonds, the expected total return is: E[Return] = (1,000,000 - x) * (1.035)^5 + x * (1 + 0.25) But since x = 1,000,000, the Steady Bonds part is zero, so E[Return] = 1,000,000 * 1.25 = 1,250,000. Alternatively, if she had invested some amount in both, the expected return would be a weighted average. But since she can invest all in Practical Ventures, the expected return is higher. Wait, but is the expected return from Practical Ventures actually 25%? Because the mean return is 25%, yes. So, the expected value is 1,000,000 * 1.25 = 1,250,000. But let me make sure. The expected return from the Practical Ventures Fund is 25%, so the expected amount is 1,000,000 * 1.25 = 1,250,000. Therefore, the total expected return is 1,250,000. But wait, if she had invested all in Steady Bonds, the return would be 1,000,000 * (1.035)^5 ≈ 1,187,686. So, investing all in Practical Ventures gives a higher expected return. Therefore, the answers are: 1. Maximum x is 1,000,000. 2. Total expected return is 1,250,000. But let me double-check the Z-score calculation. Z = (3.5 - 25)/4.472 ≈ -4.806. Looking up Z = -4.806, the cumulative probability is indeed extremely low, much less than 10%. So, the probability is already below 10%, so she can invest the entire amount. Alternatively, if the problem had meant the probability that the total return is less than 3.5%, that would be a different story. But as per the problem statement, it's about the Practical Ventures Fund's return. Therefore, I think my answers are correct. But just to be thorough, let's consider if the problem had meant the total return. If she invests x in Practical Ventures and (1,000,000 - x) in Steady Bonds, the total return would be: Return = (1,000,000 - x)*(1.035)^5 + x*(1 + R), where R ~ N(25%, 4.472%). She wants P(Return < 3.5%) ≤10%. But that's a different problem. The original problem says: "the probability of the Practical Ventures Fund yielding a return less than 3.5% over the 5-year period being no more than 10%." So, it's specifically about the Practical Ventures Fund's return, not the total return. Therefore, my initial conclusion stands. So, final answers: 1. x = 1,000,000 2. Total expected return = 1,250,000 But wait, let me calculate the exact expected return from Steady Bonds if she had invested part of the money there. But since she's investing all in Practical Ventures, the Steady Bonds part is zero. Alternatively, if she had invested some amount in both, the expected return would be a combination. But since she can invest all in Practical Ventures, the expected return is maximized. Therefore, I think my answers are correct.</think>

🧠:<think>Okay, so I have this problem about a documentary on climate change, and it's divided into two parts. Let me try to tackle them one by one.Starting with part 1: We have a function for the global average temperature, T(t) = 0.02t + 15. Then, the sea level rise S(t) is modeled by the differential equation dS/dt = 0.1(T(t) - 14). We're told that S(0) = 0, and we need to find S(t) for t from 0 to 50.Alright, so first, let me write down what I know. The differential equation is dS/dt = 0.1(T(t) - 14). Since T(t) is given as 0.02t + 15, I can substitute that into the equation.So substituting T(t) into the differential equation gives:dS/dt = 0.1((0.02t + 15) - 14)Let me compute the expression inside the parentheses first:(0.02t + 15) - 14 = 0.02t + 1So now the differential equation becomes:dS/dt = 0.1(0.02t + 1)Let me compute 0.1 times each term:0.1 * 0.02t = 0.002t0.1 * 1 = 0.1So, dS/dt = 0.002t + 0.1Now, to find S(t), I need to integrate dS/dt with respect to t.So, integrating both sides:∫ dS/dt dt = ∫ (0.002t + 0.1) dtWhich simplifies to:S(t) = ∫ (0.002t + 0.1) dtLet me compute the integral term by term.First, the integral of 0.002t with respect to t is:0.002 * (t^2)/2 = 0.001t^2Second, the integral of 0.1 with respect to t is:0.1tSo putting it together:S(t) = 0.001t^2 + 0.1t + CWhere C is the constant of integration. Now, we can use the initial condition S(0) = 0 to find C.Plugging t = 0 into S(t):S(0) = 0.001*(0)^2 + 0.1*(0) + C = 0 + 0 + C = CBut S(0) is given as 0, so C = 0.Therefore, the function S(t) is:S(t) = 0.001t^2 + 0.1tLet me double-check my steps. I substituted T(t) into the differential equation correctly, simplified the expression, integrated term by term, and applied the initial condition. Seems solid.Moving on to part 2: The population of a coastal city is modeled by P(t) = P0 e^{-kt}, where P0 is the current population. It's expected to decrease by 20% over the next 10 years. We need to find k, and then use it to calculate the population after 50 years.First, let's parse the information. A decrease of 20% over 10 years means that after 10 years, the population is 80% of the original. So, P(10) = P0 * 0.8.Given the model P(t) = P0 e^{-kt}, so plugging t = 10:P(10) = P0 e^{-10k} = 0.8 P0Divide both sides by P0:e^{-10k} = 0.8Now, take the natural logarithm of both sides:ln(e^{-10k}) = ln(0.8)Simplify left side:-10k = ln(0.8)Therefore, k = - (ln(0.8))/10Compute ln(0.8). Let me recall that ln(0.8) is negative because 0.8 is less than 1. Let me calculate it:ln(0.8) ≈ -0.22314So, k ≈ - (-0.22314)/10 ≈ 0.022314So, k ≈ 0.0223 per year.Now, we need to calculate the population after 50 years. So, P(50) = P0 e^{-k*50}Plugging in k ≈ 0.0223:P(50) = P0 e^{-0.0223*50}Compute the exponent:0.0223 * 50 = 1.115So, P(50) = P0 e^{-1.115}Calculate e^{-1.115}. Let me recall that e^{-1} ≈ 0.3679, e^{-1.1} ≈ 0.3329, e^{-1.115} is a bit less than that.Using a calculator, e^{-1.115} ≈ 0.328So, P(50) ≈ P0 * 0.328Therefore, the population is expected to decrease to approximately 32.8% of its current level after 50 years.Let me verify the steps. We used the given decrease to set up the equation, solved for k, then used that k to find the population at t=50. That seems correct.Wait, let me double-check the calculation of k.We had e^{-10k} = 0.8So, taking natural logs:-10k = ln(0.8)Therefore, k = - ln(0.8)/10Compute ln(0.8):ln(0.8) = ln(4/5) = ln(4) - ln(5) ≈ 1.3863 - 1.6094 ≈ -0.2231So, k = - (-0.2231)/10 ≈ 0.02231, which is approximately 0.0223. Correct.Then, for t=50:-kt = -0.0223*50 = -1.115e^{-1.115} ≈ 0.328. So, 32.8% of P0. That seems right.Alternatively, using more precise calculation:Compute e^{-1.115}:We know that e^{-1} ≈ 0.3678794412e^{-1.1} ≈ 0.3328710678e^{-1.115} is between these two.Compute 1.115 - 1.1 = 0.015So, the difference between e^{-1.1} and e^{-1.115} is approximately the derivative at some point times 0.015.The derivative of e^{-x} is -e^{-x}. So, the approximate decrease is -e^{-1.1} * 0.015 ≈ -0.332871 * 0.015 ≈ -0.004993Therefore, e^{-1.115} ≈ e^{-1.1} - 0.004993 ≈ 0.332871 - 0.004993 ≈ 0.327878Which is approximately 0.3279, so 32.79%, which rounds to 32.8%. So, that seems accurate.Therefore, the projected population is about 32.8% of the current population after 50 years.So, summarizing:1. The sea level function is S(t) = 0.001t² + 0.1t meters.2. The decay constant k is approximately 0.0223 per year, leading to a population of about 32.8% of the current population after 50 years.I think that's it. I don't see any mistakes in the calculations.Final Answer1. The function ( S(t) ) is boxed{0.001t^2 + 0.1t} meters.2. The projected population of the city 50 years from now is approximately boxed{32.8%} of the current population.</think>

🧠:<think>Okay, so I have this problem about Jimmy Osmond's fan event. There are two parts. Let me try to tackle them one by one.Starting with part 1: Determine the possible durations of each song if the sum of the squares of the durations is minimized. Also, I need to prove that this set is unique.Hmm, so we have 10 songs, each with a distinct positive integer duration, and the total duration is 90 minutes. We need to assign durations such that the sum of their squares is as small as possible. And then show that this set is unique.I remember that for a given sum, the sum of squares is minimized when the numbers are as equal as possible. Since the durations have to be distinct positive integers, we can't have them all the same, but we can make them as close to each other as possible.So, let me think. If all durations were equal, each would be 9 minutes because 90 divided by 10 is 9. But since they have to be distinct, we need to distribute the extra minutes around 9.Wait, actually, 10 distinct positive integers adding up to 90. So, the minimal sum of squares occurs when the numbers are consecutive integers as close to the average as possible.What's the average? 90 divided by 10 is 9. So, the numbers should be around 9. Let me try to find 10 consecutive integers centered around 9.But 10 consecutive integers would be from 5 to 14, right? Let me check: 5+6+7+8+9+10+11+12+13+14. Let's compute that.5+14=19, 6+13=19, 7+12=19, 8+11=19, 9+10=19. So, 5 times 19 is 95. But we need the total to be 90, not 95. So that's too much.Hmm, so 10 consecutive integers starting at 5 sum to 95, which is 5 more than 90. So, we need to reduce the total by 5.How can we adjust the numbers to make the total 90? Since they have to be distinct, we can't just subtract 1 from each number because that would make some numbers equal.Maybe we can subtract 1 from the largest numbers. Let's see: starting from 5 to 14, which is 10 numbers. If we subtract 1 from the last five numbers, each of them would decrease by 1, so total decrease would be 5. That would bring the total from 95 to 90.So, the numbers would be: 5,6,7,8,9,9,10,11,12,13. Wait, but 9 is repeated. That's not allowed because they have to be distinct. So, that approach doesn't work.Alternatively, maybe we can subtract 1 from the largest number and adjust others. Let me think.If I have the sequence 5 to 14, which is 10 numbers, sum 95. I need to reduce by 5. Let's try subtracting 1 from the five largest numbers, but that causes duplicates. Instead, maybe subtract 1 from the largest number and 1 from the next largest, but that still might cause duplicates.Wait, maybe a better approach is to find 10 distinct integers that sum to 90, as close as possible to each other.The minimal sum of squares occurs when the numbers are as equal as possible. So, let's try to find 10 distinct integers around 9.Let me list numbers starting from 1 upwards, but ensuring they are distinct and sum to 90.Wait, but starting from 1 would make the numbers too spread out. Maybe starting from a higher number.Alternatively, perhaps starting from 4.Let me try: 4,5,6,7,8,9,10,11,12,13. Let's sum these.4+5=9, 6+7=13, 8+9=17, 10+11=21, 12+13=25. So, 9+13=22, 22+17=39, 39+21=60, 60+25=85. So total is 85, which is 5 less than 90.So, we need to add 5 more. How can we distribute this without making duplicates.We can add 1 to the largest numbers. So, add 1 to 13, making it 14, but then we have 14 and 13, which is fine. But we still need to add 4 more.Wait, let's think differently. Maybe shift the numbers up.Alternatively, let's take the sequence 5,6,7,8,9,10,11,12,13,14, which sums to 95. We need to reduce by 5. So, we can subtract 1 from five of the numbers, but ensuring they remain distinct.If we subtract 1 from the five largest numbers: 14 becomes 13, 13 becomes 12, 12 becomes 11, 11 becomes 10, 10 becomes 9. But now we have duplicates: 9 appears twice, 10 appears twice, etc. So that's not allowed.Alternatively, subtract 1 from the five smallest numbers: 5 becomes 4, 6 becomes 5, 7 becomes 6, 8 becomes 7, 9 becomes 8. Now, the numbers are 4,5,6,7,8,10,11,12,13,14. Let's check the sum: 4+5+6+7+8+10+11+12+13+14.Compute step by step: 4+5=9, 9+6=15, 15+7=22, 22+8=30, 30+10=40, 40+11=51, 51+12=63, 63+13=76, 76+14=90. Perfect, the sum is 90.And all numbers are distinct: 4,5,6,7,8,10,11,12,13,14.So, this set sums to 90, has 10 distinct integers, and is as close as possible to each other, which should minimize the sum of squares.Is this the unique set? Let's see.Suppose there's another set of 10 distinct integers summing to 90 with a smaller sum of squares. But since we've made the numbers as close as possible, any other arrangement would have more spread out numbers, leading to a larger sum of squares.Wait, let me think. Suppose instead of subtracting 1 from the five smallest, we subtract 1 from some other numbers. For example, subtract 1 from 5,6,7,8,9, making them 4,5,6,7,8, and leave the rest as 10,11,12,13,14. That's the same as above.Alternatively, could we subtract 1 from some other combination? For example, subtract 1 from 5,6,7,8,10, making them 4,5,6,7,9, and leave the rest as 9,11,12,13,14. Wait, but 9 appears twice now, which is not allowed.So, the only way to subtract 5 without causing duplicates is to subtract 1 from the five smallest numbers, resulting in 4,5,6,7,8,10,11,12,13,14.Therefore, this set is unique because any other adjustment would either cause duplicates or not reduce the sum by exactly 5.So, the possible durations are 4,5,6,7,8,10,11,12,13,14 minutes.Now, moving on to part 2: If the event organizers want to distribute 500 limited edition memorabilia evenly among attendees such that each attendee receives an equal number of items and no items are left over, what's the maximum number of attendees possible? Also, use the greatest common divisor (GCD) of the song durations to help solve this.Hmm, so we need to find the maximum number of attendees such that 500 is divisible by that number. So, the maximum number of attendees is the largest divisor of 500.But wait, the problem says to use the GCD of the song durations. So, first, I need to find the GCD of the song durations from part 1.From part 1, the song durations are 4,5,6,7,8,10,11,12,13,14.Let me compute the GCD of these numbers.First, list the numbers: 4,5,6,7,8,10,11,12,13,14.Compute GCD step by step.Start with GCD(4,5). GCD(4,5)=1.Then GCD(1,6)=1.GCD(1,7)=1.GCD(1,8)=1.GCD(1,10)=1.GCD(1,11)=1.GCD(1,12)=1.GCD(1,13)=1.GCD(1,14)=1.So, the GCD of all song durations is 1.Therefore, the GCD is 1.Now, to find the maximum number of attendees, we need the largest divisor of 500. Since 500 divided by 1 is 500, but that would mean each attendee gets 1 item, which is possible, but maybe the problem wants the maximum number of attendees such that each gets more than one item? Or perhaps not, since it just says "an equal number of items" without specifying.Wait, the problem says "distribute 500 limited edition memorabilia evenly among attendees such that each attendee receives an equal number of items and no items are left over." So, the number of attendees must be a divisor of 500.The maximum number of attendees is the largest divisor of 500, which is 500 itself. But that would mean each attendee gets 1 item. Alternatively, if we consider that each attendee should get at least one item, then 500 is the maximum.But maybe the problem is implying that the number of attendees should be related to the GCD of the song durations, which is 1. Since the GCD is 1, the maximum number of attendees is 500, because 500 divided by 1 is 500.Wait, but perhaps I'm overcomplicating. The GCD is 1, so the maximum number of attendees is 500, as 500 is divisible by 1.Alternatively, maybe the problem is suggesting that the number of attendees should be a divisor of both 500 and the GCD of the song durations. But since the GCD is 1, the only common divisor is 1, so the maximum number of attendees is 500.Wait, but let me think again. The problem says "use the greatest common divisor of the song durations to help solve the problem." So, perhaps the number of attendees must be a divisor of both 500 and the GCD of the song durations. But since the GCD is 1, the only possible number is 1, which doesn't make sense because you can't have 1 attendee if you want to distribute 500 items. Alternatively, maybe the number of attendees must be a divisor of 500 and also a multiple of the GCD, but since GCD is 1, any divisor of 500 is acceptable. Therefore, the maximum number of attendees is 500.But that seems a bit trivial. Maybe I'm missing something. Let me check the problem again."If the event organizers want to distribute 500 limited edition Jimmy Osmond memorabilia evenly among attendees such that each attendee receives an equal number of items and no items are left over, what is the maximum number of attendees possible? Determine the greatest common divisor of the song durations and use this to help solve the problem."So, the key is to use the GCD of the song durations. Since the GCD is 1, the maximum number of attendees is 500 because 500 is divisible by 1. But that seems too straightforward. Alternatively, maybe the number of attendees must be a divisor of 500 and also a divisor of the GCD, but since GCD is 1, the only possible number is 1, which doesn't make sense. Therefore, perhaps the maximum number of attendees is 500.Wait, but maybe the problem is implying that the number of attendees should be a divisor of both 500 and the GCD of the song durations. But since GCD is 1, the only common divisor is 1, so the maximum number of attendees is 1, which can't be right because you can't have 1 attendee if you want to distribute 500 items. So, perhaps the problem is just asking for the maximum number of attendees, which is 500, regardless of the GCD, but the GCD is 1, so it's just 500.Alternatively, maybe the problem is suggesting that the number of attendees should be a divisor of 500 and also a divisor of the sum of the song durations, which is 90. But that's not what the problem says. It says to use the GCD of the song durations.Wait, perhaps the problem is implying that the number of attendees should be a divisor of both 500 and the GCD of the song durations. Since GCD is 1, the maximum number is 1, but that doesn't make sense. Alternatively, maybe the number of attendees should be a divisor of 500 and also a multiple of the GCD, which is 1, so any divisor of 500 is acceptable, making the maximum 500.I think that's the case. So, the maximum number of attendees is 500.But let me double-check. The GCD of the song durations is 1, so the maximum number of attendees is 500 because 500 is divisible by 1. Therefore, each attendee would receive 1 item.Alternatively, if the problem requires each attendee to receive more than one item, then the maximum number of attendees would be 250, because 500 divided by 2 is 250, but that's not necessarily related to the GCD. Since the GCD is 1, the maximum number is 500.I think that's the answer.</think>

🧠:<think>Okay, so I have this problem where a weather expert is looking at the correlation between global temperature anomalies and the frequency of extreme weather events over the past 50 years. They've given me two tasks: first, to perform a linear regression analysis to find the best-fit line, and second, to fit a quadratic regression model and compare the goodness-of-fit between the linear and quadratic models.Alright, let me start by understanding what linear regression is. From what I remember, linear regression is a statistical method that models the relationship between a dependent variable and one or more independent variables. In this case, the dependent variable is the frequency of extreme weather events, and the independent variable is the global temperature anomaly.So, the goal is to find the best-fit line that describes this relationship. The equation of a line is usually written as y = mx + b, where m is the slope and b is the y-intercept. In regression terms, this would translate to Frequency = β1 * Temperature + β0, where β1 is the slope and β0 is the y-intercept.To find the best-fit line, we need to calculate the coefficients β0 and β1. I think this is done by minimizing the sum of the squared differences between the observed values and the values predicted by the line. This method is called the least squares method.But wait, how exactly do we calculate β1 and β0? I recall there are formulas for these. Let me try to remember. The slope β1 can be calculated using the covariance of the two variables divided by the variance of the independent variable. The formula is something like:β1 = Cov(X, Y) / Var(X)And the y-intercept β0 is calculated as:β0 = ȳ - β1 * x̄Where ȳ is the mean of the dependent variable (frequency) and x̄ is the mean of the independent variable (temperature).So, to compute these, I would need the means of both variables, their covariance, and the variance of the temperature anomalies.But hold on, I don't have the actual dataset. The problem statement doesn't provide specific numbers, so maybe I need to outline the steps rather than compute exact values? Or perhaps I should assume some hypothetical data? Hmm, the question says "using the dataset," but since I don't have access to it, maybe I should explain the process.Alright, so assuming I have the dataset, here's what I would do step by step:1. Data Preparation: I would start by loading the dataset, which contains two columns: one for the yearly global temperature anomalies and another for the number of extreme weather events each year.2. Calculate Means: Compute the mean of the temperature anomalies (x̄) and the mean of the frequency of extreme weather events (ȳ).3. Compute Covariance and Variance: Calculate the covariance between temperature anomalies and frequency, and the variance of temperature anomalies.4. Calculate Slope (β1): Using the formula Cov(X, Y) / Var(X).5. Calculate Intercept (β0): Using ȳ - β1 * x̄.6. Formulate the Regression Equation: Once I have β0 and β1, the equation would be Frequency = β1 * Temperature + β0.7. Evaluate the Model: To assess how well the model fits the data, I would calculate the R-squared value. R-squared represents the proportion of variance in the dependent variable that's explained by the independent variable. A higher R-squared value indicates a better fit.Now, moving on to the second part: fitting a quadratic regression model. A quadratic model is a type of polynomial regression where the relationship between the variables is modeled as a quadratic equation. The general form is:Frequency = β2 * Temperature² + β1 * Temperature + β0Here, β2 is the coefficient for the squared term, β1 is the coefficient for the linear term, and β0 is the constant term.To fit this model, I would need to use a method that can handle multiple terms, like the method of least squares again, but extended to multiple variables. This might involve setting up a system of equations or using matrix algebra to solve for the coefficients.Alternatively, many statistical software packages and programming languages (like Python or R) have built-in functions to perform polynomial regression. For example, in Python, using the numpy library, you can use the polyfit function which can fit a polynomial of a specified degree to the data.Once I have the coefficients for the quadratic model, I would again calculate the R-squared value to compare it with the linear model. If the quadratic model has a higher R-squared value, it suggests that the quadratic relationship explains more variance in the data than the linear model. However, I should also be cautious about overfitting, where a more complex model (like quadratic) might fit the noise in the data rather than the underlying relationship.Another thing to consider is the significance of the coefficients. Even if the R-squared is higher for the quadratic model, the additional terms (like the squared term) should be statistically significant. This can be checked using hypothesis tests or p-values.I also remember that sometimes, when you add more terms to a model, the R-squared will always increase, but the adjusted R-squared accounts for the number of terms in the model. So, maybe comparing adjusted R-squared values would be more appropriate to see if the additional complexity of the quadratic model is justified.But again, without the actual data, I can't compute these values. So, perhaps I should explain the process and the reasoning behind choosing one model over the other.In summary, for the first task, I would perform a linear regression by calculating the slope and intercept using the means, covariance, and variance. For the second task, I would fit a quadratic model, likely using a more advanced method or software, and then compare the goodness-of-fit using R-squared or adjusted R-squared values to determine which model is better.I should also consider visualizing the data. Plotting the temperature anomalies against the frequency of extreme weather events could give a visual indication of whether a linear or quadratic relationship is more appropriate. If the data points seem to follow a curve, the quadratic model might be a better fit. If they seem to follow a straight line, the linear model might suffice.Additionally, residual analysis could be useful. After fitting both models, I could plot the residuals (the difference between observed and predicted values) to check for patterns. If the residuals from the linear model show a pattern that could be explained by a quadratic term, that might suggest that the quadratic model is more appropriate.I also wonder about the units and scales of the variables. Temperature anomalies are in degrees Celsius, and the frequency is a count of events. It's important to ensure that the variables are appropriately scaled and that transformations aren't needed. For example, sometimes taking the logarithm of the dependent variable can help, but in this case, since we're dealing with counts, maybe a Poisson regression or another generalized linear model would be more appropriate. However, since the question specifies linear and quadratic regression, I think we can stick with those.Another consideration is the possibility of autocorrelation, especially since the data is time-series. Extreme weather events in one year might be related to those in the previous year, and similarly for temperature anomalies. However, since the question doesn't mention this, I might not need to address it unless it's part of the analysis.In terms of hypothesis testing, I could test whether the quadratic term is statistically significant. If the p-value for β2 is low (e.g., less than 0.05), it suggests that the quadratic term contributes significantly to the model, supporting the use of the quadratic model over the linear one.I should also think about the practical significance. Even if the quadratic model has a slightly higher R-squared, the difference might not be meaningful in a real-world context. It's important to consider both statistical and practical significance.Lastly, I might want to check for any outliers or influential points in the dataset that could disproportionately affect the regression results. Outliers can have a large impact on the slope and intercept of the regression line, so identifying and handling them appropriately is crucial.Overall, while I can't compute the exact coefficients without the data, I can outline the steps and considerations for performing both linear and quadratic regression analyses and comparing their goodness-of-fit.</think>