🧠:<think>Alright, so I have this problem about vowel harmony in a hypothetical Southeast Asian language. The language has 6 distinct vowels, each assigned a unique numerical value from 1 to 6. The first part of the problem asks me to determine the number of distinct vowel pairs (where order matters) that can occur in a single word when the sum of their numerical values is a constant ( k = 7 ). The words can have up to 4 vowels, and vowels can repeat. Okay, let's break this down. First, I need to find all possible ordered pairs of vowels (a, b) such that a + b = 7. Since each vowel is represented by a number from 1 to 6, I can list out all possible pairs where the sum is 7.Starting with the first vowel as 1: 1 + 6 = 7, so (1,6) is a pair.Next, first vowel as 2: 2 + 5 = 7, so (2,5).First vowel as 3: 3 + 4 = 7, so (3,4).First vowel as 4: 4 + 3 = 7, so (4,3).First vowel as 5: 5 + 2 = 7, so (5,2).First vowel as 6: 6 + 1 = 7, so (6,1).So, listing them out, the pairs are (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). That's 6 pairs in total.Wait, but the problem mentions that the language allows for words with up to 4 vowels. Does that affect the number of vowel pairs? Hmm, I think not directly, because we're just looking for pairs within a word, regardless of the word's length. So whether the word has 2, 3, or 4 vowels, any two vowels in sequence can form a pair. But since the question is about the number of distinct vowel pairs (order matters) that can occur in a single word, I think it's just about the possible ordered pairs, regardless of the word length. So, the number is 6.But hold on, the problem says "vowel pairs occurring in a word such that the sum of their numerical values is a constant ( k = 7 )". So, regardless of the word's length, as long as two vowels in the word sum to 7, they form a pair. But since the word can have up to 4 vowels, how does that influence the number of pairs? Wait, maybe I'm overcomplicating. The question is asking for the number of distinct vowel pairs (order matters) that can occur in a single word when ( k = 7 ). So, it's not about how many pairs are in a word, but how many possible distinct pairs exist where the sum is 7. So, as I listed before, it's 6.But let me think again. Since order matters, each pair is unique based on the order, so (1,6) is different from (6,1), etc. So, as I listed, 6 pairs. So, the answer is 6.Wait, but hold on. If the word can have up to 4 vowels, does that mean that the same vowel can appear multiple times? The problem says vowels can repeat in a word, so yes, repetition is allowed. But in terms of pairs, even if a vowel is repeated, the pair (a, a) would only be possible if 2a = 7, which is not possible because 7 is odd and a is an integer from 1 to 6. So, no such pairs where a = b. So, all pairs must be distinct in terms of their numerical values, but order matters.Therefore, the number of distinct ordered pairs where a + b = 7 is 6. So, the answer is 6.Moving on to the second part. The enthusiast observes that certain vowel pairs are more frequent due to cultural influences. The frequency of a vowel pair (a, b) is inversely proportional to the product of their numerical values in a word of 4 vowels. The total word frequency is normalized to 1, and we need to derive an expression for the expected frequency of the vowel pair (a, b) in a randomly selected word, given that ( a + b = 7 ).Alright, so first, let's parse this. The frequency of a vowel pair (a, b) is inversely proportional to the product of their numerical values. So, if the product is larger, the frequency is smaller, and vice versa.Given that the word has 4 vowels, how does this affect the frequency? Hmm, the word has 4 vowels, so there are 3 possible adjacent pairs in a 4-vowel word. For example, in a word with vowels w1, w2, w3, w4, the pairs are (w1, w2), (w2, w3), (w3, w4). So, each word contributes 3 pairs.But the problem says the frequency of a vowel pair (a, b) is inversely proportional to the product of their numerical values. So, for each occurrence of (a, b) in any word, its frequency is proportional to 1/(a*b). But since the total word frequency is normalized to 1, we need to compute the expected frequency.Wait, perhaps I need to model this as a probability distribution. Let me think.First, all possible 4-vowel words are considered. Each word is a sequence of 4 vowels, each from 1 to 6, with repetition allowed. So, the total number of possible words is 6^4 = 1296.Each word has 3 adjacent vowel pairs. So, the total number of vowel pairs across all words is 3 * 1296 = 3888.But the frequency of each vowel pair (a, b) is inversely proportional to a*b. So, the frequency of (a, b) is proportional to 1/(a*b). Therefore, the probability of a word containing the pair (a, b) is proportional to 1/(a*b). But wait, actually, each word contributes multiple pairs, so the total contribution of a word is the sum over its pairs of 1/(a*b). But since the total word frequency is normalized to 1, we need to compute the expected frequency.Wait, perhaps it's better to think in terms of the expected number of times a pair (a, b) appears in a word. But the problem says the frequency of the pair is inversely proportional to the product of their numerical values. So, perhaps the probability of the pair (a, b) is proportional to 1/(a*b), and since the total probability is 1, we can normalize accordingly.But let me try to formalize this.Let’s denote the frequency of pair (a, b) as f(a, b). According to the problem, f(a, b) is inversely proportional to a*b, so f(a, b) = C / (a*b), where C is the constant of proportionality.Since the total frequency is normalized to 1, the sum over all possible pairs (a, b) of f(a, b) should equal 1.But wait, in a word of 4 vowels, each word contributes 3 pairs. So, the total number of pairs across all words is 3 * 6^4 = 3888. But the frequency of each pair is inversely proportional to a*b, so the total sum over all pairs is sum_{a=1 to 6} sum_{b=1 to 6} (C / (a*b)) = C * (sum_{a=1 to 6} 1/a) * (sum_{b=1 to 6} 1/b) = C * (H_6)^2, where H_6 is the 6th harmonic number.But wait, actually, in the context of the problem, the frequency is per word. Each word contributes 3 pairs, each with their own frequency. So, the total frequency across all words is the sum over all words of the sum over their pairs of f(a, b). But since each pair's frequency is inversely proportional to a*b, and the total word frequency is normalized to 1, perhaps we need to compute the expected frequency of a specific pair (a, b) given that a + b = 7.Wait, the problem says: "derive an expression for the expected frequency of the vowel pair (a, b) in a randomly selected word, given that ( a + b = 7 )."So, given that a + b = 7, we need to find the expected frequency of (a, b). So, first, we have the pairs (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). Each of these pairs has a frequency inversely proportional to a*b. So, for each pair (a, b), f(a, b) = C / (a*b). Since the total frequency is 1, the sum of f(a, b) over all pairs (a, b) where a + b = 7 should equal 1.Wait, no. The total word frequency is normalized to 1, but each word contributes multiple pairs. So, perhaps the total frequency across all pairs is more than 1. Wait, I'm getting confused.Let me think differently. The frequency of a pair (a, b) is inversely proportional to a*b. So, the probability of a pair (a, b) occurring in a word is proportional to 1/(a*b). But since each word has 3 pairs, the total probability across all pairs in all words would be 3 * 1 = 3, but the problem says the total word frequency is normalized to 1. Hmm, maybe I need to adjust.Alternatively, perhaps the frequency of each pair is considered in the context of all possible pairs in all words. So, the total number of pairs is 3 * 6^4 = 3888. Each pair (a, b) occurs in some number of words. The frequency of (a, b) is the number of times it appears across all words, divided by the total number of pairs (3888). But the problem says the frequency is inversely proportional to a*b. So, the number of times (a, b) appears is proportional to 1/(a*b). Therefore, the frequency f(a, b) = (number of occurrences of (a, b)) / 3888 = C / (a*b), where C is a constant such that the sum over all pairs of f(a, b) = 1.But wait, the sum over all pairs of f(a, b) would be sum_{a=1 to 6} sum_{b=1 to 6} (C / (a*b)) = C * (sum_{a=1 to 6} 1/a) * (sum_{b=1 to 6} 1/b) = C * (H_6)^2, where H_6 is the 6th harmonic number. H_6 = 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 ≈ 2.45.So, sum f(a, b) = C * (2.45)^2 ≈ C * 6.0025. But since the total frequency is 1, we have C * 6.0025 = 1, so C ≈ 1/6.0025 ≈ 0.1666.But wait, the problem is specifically about pairs where a + b = 7. So, we need to consider only those pairs. So, the total frequency for these pairs would be sum_{(a,b): a+b=7} f(a, b) = sum_{(a,b): a+b=7} (C / (a*b)).But since the total word frequency is normalized to 1, does that mean the sum over all pairs (including those not summing to 7) is 1? Or is the normalization only for the pairs that sum to 7? The problem says "the total word frequency is normalized to 1", so I think it refers to the overall frequency across all possible pairs, not just those summing to 7.But the question is asking for the expected frequency of the vowel pair (a, b) in a randomly selected word, given that a + b = 7. So, perhaps we need to compute the probability of (a, b) given that a + b = 7.Wait, that might be a different approach. If we condition on a + b = 7, then we can compute the expected frequency as the probability of (a, b) given that a + b = 7.So, first, the probability of a pair (a, b) is proportional to 1/(a*b). So, the probability P(a, b) = C / (a*b), where C is the normalization constant such that sum_{(a,b): a + b =7} P(a, b) = 1.So, the normalization constant C is 1 divided by the sum over all (a, b) with a + b =7 of 1/(a*b).From part 1, we have the pairs: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). So, let's compute the sum:For (1,6): 1/(1*6) = 1/6 ≈ 0.1667For (2,5): 1/(2*5) = 1/10 = 0.1For (3,4): 1/(3*4) = 1/12 ≈ 0.0833For (4,3): 1/(4*3) = 1/12 ≈ 0.0833For (5,2): 1/(5*2) = 1/10 = 0.1For (6,1): 1/(6*1) = 1/6 ≈ 0.1667Adding these up: 0.1667 + 0.1 + 0.0833 + 0.0833 + 0.1 + 0.1667 ≈ Let's compute:0.1667 + 0.1 = 0.26670.2667 + 0.0833 = 0.350.35 + 0.0833 = 0.43330.4333 + 0.1 = 0.53330.5333 + 0.1667 = 0.7So, the total sum is 0.7. Therefore, the normalization constant C is 1 / 0.7 ≈ 1.4286.Therefore, the probability P(a, b) = (1 / (a*b)) / 0.7 = (1 / (a*b)) * (10/7) ≈ (1.4286) / (a*b).So, the expected frequency of the vowel pair (a, b) given that a + b =7 is P(a, b) = (1 / (a*b)) / 0.7 = 10/(7*a*b).Therefore, the expression is 10/(7*a*b).Wait, let me verify:Sum over all pairs (a,b) with a + b =7 of P(a, b) = sum (10/(7*a*b)) = 10/7 * sum (1/(a*b)) = 10/7 * 0.7 = 10/7 * 7/10 = 1. So, yes, that works.Therefore, the expected frequency is 10/(7*a*b).So, the expression is 10/(7ab).But let me write it as 10/(7ab) or (10)/(7ab).Alternatively, since 10/7 is approximately 1.4286, but we can keep it as a fraction.So, the expected frequency is (10)/(7ab).Therefore, the answer is 10/(7ab).Wait, but let me think again. The problem says "the frequency of a vowel pair (a, b) is inversely proportional to the product of their numerical values in a word of 4 vowels." So, is the frequency per word or per pair?Wait, perhaps I need to model it differently. Each word has 3 pairs, and each pair contributes to the frequency. So, the total frequency across all words is the sum over all words of the sum over their pairs of f(a, b). But since each word is equally likely, and the total word frequency is normalized to 1, the expected frequency of a pair (a, b) is the number of times (a, b) appears across all words divided by the total number of pairs across all words.But the problem states that the frequency is inversely proportional to a*b. So, perhaps the number of times (a, b) appears is proportional to 1/(a*b). Therefore, the expected frequency is (number of occurrences of (a, b)) / (total number of pairs) = (C / (a*b)) / (3 * 6^4).But since the total word frequency is normalized to 1, perhaps the expected frequency is just proportional to 1/(a*b), normalized over all pairs.Wait, I think my initial approach was correct. Since the frequency is inversely proportional to a*b, and given that a + b =7, the probability is (1/(a*b)) divided by the sum over all such pairs of 1/(a*b), which is 0.7. So, the expected frequency is (1/(a*b)) / 0.7 = 10/(7ab).Yes, that seems correct.So, summarizing:1. The number of distinct vowel pairs (order matters) with sum 7 is 6.2. The expected frequency of a pair (a, b) given a + b =7 is 10/(7ab).Therefore, the answers are 6 and 10/(7ab).</think>

🧠:<think>Alright, so I have this problem about optimizing animations on a webpage. Let me try to break it down step by step.First, the YouTuber has a sequence of n animations. Each animation has a duration d_i and a delay t_i. The total time T for all animations to complete is the maximum of (d_i + t_i) across all i. So, T is the longest time any single animation takes, considering both its duration and delay.The constraints are that the sum of all durations is D, and the sum of all delays is T_d. So, we have:Sum of d_i from i=1 to n = DSum of t_i from i=1 to n = T_dOur goal is to find the values of d_i and t_i that minimize T, which is max(d_i + t_i).Hmm, okay. So, to minimize the maximum of d_i + t_i, given that the total durations and delays are fixed.Let me think about how to approach this. It seems like an optimization problem where we need to distribute D and T_d across n animations such that the maximum (d_i + t_i) is as small as possible.I remember that in optimization, when you want to minimize the maximum of something, it's often related to equalizing the values. So, maybe if we can make all d_i + t_i equal, that would give the minimal maximum.Let me test this idea. Suppose all d_i + t_i = T. Then, since we have n animations, the sum of (d_i + t_i) would be n*T.But the sum of d_i is D, and the sum of t_i is T_d, so the total sum is D + T_d. Therefore, n*T = D + T_d, which gives T = (D + T_d)/n.So, if we can set each d_i + t_i equal to (D + T_d)/n, then the maximum T would be minimized.But wait, is this always possible? Let's see.We have two constraints:1. Sum(d_i) = D2. Sum(t_i) = T_dIf we set each d_i + t_i = T, then for each i, t_i = T - d_i.Substituting into the second constraint:Sum(t_i) = Sum(T - d_i) = n*T - Sum(d_i) = n*T - D = T_dSo, n*T - D = T_d => n*T = D + T_d => T = (D + T_d)/nWhich is consistent with what I had before.Therefore, if we can set each d_i + t_i = (D + T_d)/n, then T is minimized.But does this mean that all d_i + t_i must be equal? Or can some be less and some be equal?Wait, since we are trying to minimize the maximum, the minimal possible maximum is when all d_i + t_i are equal. Because if one is larger, then the maximum would be larger, which is not optimal. If one is smaller, then we could potentially increase it without affecting the maximum, but that might not help.Therefore, the minimal maximum occurs when all d_i + t_i are equal, which is (D + T_d)/n.So, to achieve this, we need to set each d_i + t_i = (D + T_d)/n.But how do we distribute d_i and t_i? There are multiple ways, but perhaps the simplest is to set all d_i equal and all t_i equal.Wait, but that might not be necessary. Let me think.Suppose we set each d_i = D/n and each t_i = T_d/n. Then, d_i + t_i = (D + T_d)/n, which is exactly what we want. So, in this case, each animation has the same duration and the same delay.But is this the only way? Or can we have different d_i and t_i as long as their sum is (D + T_d)/n?Yes, as long as for each i, d_i + t_i = (D + T_d)/n, the maximum will be (D + T_d)/n. So, the distribution of d_i and t_i can vary, but their sum must be equal for all i.But since we have two variables per i, d_i and t_i, and only one equation per i (d_i + t_i = T), we have some degrees of freedom. However, the constraints on the sums of d_i and t_i must also be satisfied.Wait, if we set each d_i + t_i = T, then:Sum(d_i) = DSum(t_i) = T_dBut also, Sum(d_i + t_i) = n*T = D + T_d, which is consistent.So, as long as for each i, d_i + t_i = T, and the total sums are D and T_d, we can have different d_i and t_i.But to minimize T, we need to set T as small as possible, which is (D + T_d)/n.Therefore, the minimal T is (D + T_d)/n, achieved when each d_i + t_i = (D + T_d)/n.So, the values of d_i and t_i can be any that satisfy d_i + t_i = (D + T_d)/n for each i, with the total durations summing to D and total delays summing to T_d.But how do we choose d_i and t_i specifically?If we set all d_i equal and all t_i equal, then d_i = D/n and t_i = T_d/n, which satisfies d_i + t_i = (D + T_d)/n.Alternatively, we could have some d_i larger and some smaller, as long as their corresponding t_i adjust accordingly to keep d_i + t_i constant.But since the problem doesn't specify any other constraints on d_i and t_i, such as minimum or maximum values, the simplest solution is to set all d_i equal and all t_i equal.Therefore, the optimal values are:d_i = D/n for all it_i = T_d/n for all iThis ensures that each d_i + t_i = (D + T_d)/n, which is the minimal possible maximum T.So, for part 1, the minimal T is (D + T_d)/n, achieved by setting each d_i = D/n and each t_i = T_d/n.Now, moving on to part 2.The performance metric P is given by P = k * n / T, where k is a proportionality constant.From part 1, we have T = (D + T_d)/n.So, substituting T into P:P = k * n / [(D + T_d)/n] = k * n^2 / (D + T_d)Therefore, the maximum possible performance metric P is k * n^2 / (D + T_d).Wait, but is this the maximum? Since P is directly proportional to n and inversely proportional to T, and we have minimized T, which would maximize P.Yes, because T is minimized, so 1/T is maximized, hence P is maximized.Therefore, the maximum P is k * n^2 / (D + T_d).But let me double-check.Given that T is minimized as (D + T_d)/n, then P = k * n / T = k * n / [(D + T_d)/n] = k * n^2 / (D + T_d). Yes, that seems correct.So, the expression for the maximum possible performance metric P is k multiplied by n squared divided by (D + T_d).I think that's the answer.Final Answer1. The minimal total time ( T ) is (boxed{dfrac{D + T_d}{n}}).2. The maximum performance metric ( P ) is (boxed{k dfrac{n^2}{D + T_d}}).</think>

🧠:<think>Okay, so I have this problem about population growth using the logistic model. It's divided into two parts. Let me try to tackle them one by one.Starting with the first part: I need to find the population P(t) as a function of time t, given the logistic growth model. The differential equation is dP/dt = rP(1 - P/K). The initial population P(0) is 100, the carrying capacity K is 1000, and the growth rate r is 0.1 per year.Hmm, I remember that the logistic equation is a common model for population growth where the growth rate decreases as the population approaches the carrying capacity. The solution to this differential equation is usually given by a specific formula, but I should probably derive it to make sure I understand.The logistic equation is a separable differential equation. So, I can rewrite it as:dP / [P(1 - P/K)] = r dtTo integrate both sides, I think I need to use partial fractions on the left side. Let me set up the partial fractions:1 / [P(1 - P/K)] = A/P + B/(1 - P/K)Multiplying both sides by P(1 - P/K):1 = A(1 - P/K) + BPExpanding the right side:1 = A - (A/K)P + BPNow, grouping the terms with P:1 = A + (B - A/K)PSince this must hold for all P, the coefficients of like terms must be equal. So,For the constant term: A = 1For the P term: B - A/K = 0 => B = A/K = 1/KSo, A is 1 and B is 1/K.Therefore, the integral becomes:∫ [1/P + (1/K)/(1 - P/K)] dP = ∫ r dtLet me compute each integral separately.First integral: ∫1/P dP = ln|P| + CSecond integral: ∫(1/K)/(1 - P/K) dP. Let me make a substitution. Let u = 1 - P/K, so du = -1/K dP. Therefore, -du = (1/K) dP.So, the integral becomes ∫ (1/K) * (-du)/u = -∫ du/u = -ln|u| + C = -ln|1 - P/K| + CPutting it all together:ln|P| - ln|1 - P/K| = rt + CSimplify the left side using logarithm properties:ln|P / (1 - P/K)| = rt + CExponentiating both sides:P / (1 - P/K) = e^{rt + C} = e^C e^{rt}Let me denote e^C as another constant, say, C'.So,P / (1 - P/K) = C' e^{rt}Now, solve for P:Multiply both sides by (1 - P/K):P = C' e^{rt} (1 - P/K)Expand the right side:P = C' e^{rt} - (C' e^{rt} P)/KBring the term with P to the left side:P + (C' e^{rt} P)/K = C' e^{rt}Factor out P:P [1 + (C' e^{rt})/K] = C' e^{rt}Solve for P:P = [C' e^{rt}] / [1 + (C' e^{rt})/K]Multiply numerator and denominator by K to simplify:P = [C' K e^{rt}] / [K + C' e^{rt}]Now, apply the initial condition P(0) = 100.At t = 0, P = 100:100 = [C' K e^{0}] / [K + C' e^{0}] = [C' K] / [K + C']Multiply both sides by (K + C'):100(K + C') = C' KExpand:100K + 100C' = C' KBring all terms to one side:100K = C' K - 100C'Factor out C':100K = C'(K - 100)Solve for C':C' = (100K) / (K - 100)Given that K = 1000,C' = (100 * 1000) / (1000 - 100) = 100000 / 900 ≈ 111.111...But let's keep it exact: 100000 / 900 = 1000 / 9 ≈ 111.111...So, C' = 1000/9Therefore, plug back into the expression for P(t):P(t) = [ (1000/9) * 1000 * e^{0.1 t} ] / [1000 + (1000/9) e^{0.1 t} ]Simplify numerator and denominator:Numerator: (1000/9)*1000 = 1,000,000 / 9Denominator: 1000 + (1000/9) e^{0.1 t} = 1000(1 + (1/9) e^{0.1 t})So,P(t) = (1,000,000 / 9) e^{0.1 t} / [1000(1 + (1/9) e^{0.1 t})]Simplify:Divide numerator and denominator by 1000:P(t) = (1000 / 9) e^{0.1 t} / [1 + (1/9) e^{0.1 t}]Multiply numerator and denominator by 9 to eliminate fractions:P(t) = 1000 e^{0.1 t} / [9 + e^{0.1 t}]So, that's the expression for P(t).Wait, let me double-check the algebra:Starting from:P(t) = [C' K e^{rt}] / [K + C' e^{rt}]C' = 1000/9, K = 1000, r = 0.1So,P(t) = [ (1000/9)*1000 e^{0.1 t} ] / [1000 + (1000/9) e^{0.1 t} ]Factor numerator and denominator:Numerator: (1000/9)*1000 e^{0.1 t} = (1000^2 / 9) e^{0.1 t}Denominator: 1000 + (1000/9) e^{0.1 t} = (1000)(1 + (1/9) e^{0.1 t})So, P(t) = [ (1000^2 / 9) e^{0.1 t} ] / [1000 (1 + (1/9) e^{0.1 t}) ] = (1000 / 9) e^{0.1 t} / [1 + (1/9) e^{0.1 t} ]Multiply numerator and denominator by 9:P(t) = 1000 e^{0.1 t} / [9 + e^{0.1 t}]Yes, that's correct.Alternatively, sometimes the logistic equation is written as P(t) = K / (1 + (K/P0 - 1) e^{-rt})Let me check if this matches.Given P0 = 100, K = 1000, r = 0.1.So,P(t) = 1000 / [1 + (1000/100 - 1) e^{-0.1 t}] = 1000 / [1 + (10 - 1) e^{-0.1 t}] = 1000 / [1 + 9 e^{-0.1 t}]Hmm, that's a different expression. Wait, is that equivalent to what I have?I have P(t) = 1000 e^{0.1 t} / [9 + e^{0.1 t}]Let me manipulate the standard form:1000 / [1 + 9 e^{-0.1 t}] = 1000 e^{0.1 t} / [e^{0.1 t} + 9]Which is the same as 1000 e^{0.1 t} / [9 + e^{0.1 t}]Yes, so both expressions are equivalent. So, that's correct.Therefore, the answer to part 1 is P(t) = 1000 e^{0.1 t} / (9 + e^{0.1 t})Alternatively, it can be written as P(t) = 1000 / (1 + 9 e^{-0.1 t})Either form is acceptable, but since the question didn't specify, I think either is fine. Maybe the first form is more straightforward.Moving on to part 2: Determine the year when the population reaches 90% of the carrying capacity.90% of K is 0.9 * 1000 = 900.So, set P(t) = 900 and solve for t.Using the expression P(t) = 1000 e^{0.1 t} / (9 + e^{0.1 t})So,900 = 1000 e^{0.1 t} / (9 + e^{0.1 t})Multiply both sides by (9 + e^{0.1 t}):900 (9 + e^{0.1 t}) = 1000 e^{0.1 t}Compute 900*9 = 8100So,8100 + 900 e^{0.1 t} = 1000 e^{0.1 t}Subtract 900 e^{0.1 t} from both sides:8100 = 100 e^{0.1 t}Divide both sides by 100:81 = e^{0.1 t}Take natural logarithm of both sides:ln(81) = 0.1 tSo,t = ln(81) / 0.1Compute ln(81). Since 81 is 3^4, ln(81) = 4 ln(3). ln(3) is approximately 1.0986, so ln(81) ≈ 4 * 1.0986 ≈ 4.3944Therefore,t ≈ 4.3944 / 0.1 ≈ 43.944 yearsSo, approximately 44 years.Let me verify the steps:Starting with P(t) = 900:900 = 1000 e^{0.1 t} / (9 + e^{0.1 t})Multiply both sides by denominator:900*(9 + e^{0.1 t}) = 1000 e^{0.1 t}8100 + 900 e^{0.1 t} = 1000 e^{0.1 t}Subtract 900 e^{0.1 t}:8100 = 100 e^{0.1 t}Divide by 100:81 = e^{0.1 t}Take ln:ln(81) = 0.1 t => t = ln(81)/0.1Yes, that's correct.Alternatively, using the other form of P(t):P(t) = 1000 / (1 + 9 e^{-0.1 t})Set equal to 900:900 = 1000 / (1 + 9 e^{-0.1 t})Multiply both sides by denominator:900 (1 + 9 e^{-0.1 t}) = 1000Divide both sides by 900:1 + 9 e^{-0.1 t} = 1000 / 900 ≈ 1.1111Subtract 1:9 e^{-0.1 t} ≈ 0.1111Divide by 9:e^{-0.1 t} ≈ 0.012345679Take natural log:-0.1 t ≈ ln(0.012345679)Compute ln(0.012345679). Since 0.012345679 is approximately 1/81, ln(1/81) = -ln(81) ≈ -4.3944So,-0.1 t ≈ -4.3944Multiply both sides by -1:0.1 t ≈ 4.3944Thus,t ≈ 4.3944 / 0.1 ≈ 43.944 yearsSame result. So, that's consistent.Therefore, the population reaches 90% of the carrying capacity after approximately 43.944 years. Since the question asks for the time in years since the initial measurement, we can round it to the nearest whole number, which is 44 years.So, summarizing:1. The population function is P(t) = 1000 e^{0.1 t} / (9 + e^{0.1 t})2. The time to reach 90% of K is approximately 44 years.Final Answer1. The population as a function of time is boxed{P(t) = dfrac{1000 e^{0.1 t}}{9 + e^{0.1 t}}}.2. The population reaches 90% of the carrying capacity after boxed{44} years.</think>

🧠:<think>Alright, so Maria has this data on outreach events and community engagement, and she wants to model the relationship between them using a least-squares regression line. I need to figure out the equation of that line and then use it to determine the expected increase in engagement if the number of events is doubled from the average.First, let me recall what a least-squares regression line is. It's a line that best fits the data points by minimizing the sum of the squares of the vertical distances between the observed points and the line. The equation is of the form ( M = aE + b ), where ( a ) is the slope and ( b ) is the y-intercept.To find ( a ) and ( b ), I remember the formulas:[a = frac{nsum (E_i M_i) - sum E_i sum M_i}{nsum E_i^2 - (sum E_i)^2}][b = frac{sum M_i - a sum E_i}{n}]Where ( n ) is the number of data points, which in this case is 10.So, I need to compute several sums: the sum of ( E_i ), the sum of ( M_i ), the sum of ( E_i M_i ), and the sum of ( E_i^2 ).Let me list out the data points first:Month | E_i | M_i---|---|---1 | 2 | 502 | 3 | 703 | 5 | 1004 | 4 | 805 | 6 | 1206 | 7 | 1407 | 8 | 1608 | 6 | 1309 | 5 | 11010 | 7 | 150Now, I'll compute each required sum step by step.First, sum of ( E_i ):2 + 3 + 5 + 4 + 6 + 7 + 8 + 6 + 5 + 7Let me add them up:2 + 3 = 55 + 5 = 1010 + 4 = 1414 + 6 = 2020 + 7 = 2727 + 8 = 3535 + 6 = 4141 + 5 = 4646 + 7 = 53So, ( sum E_i = 53 )Next, sum of ( M_i ):50 + 70 + 100 + 80 + 120 + 140 + 160 + 130 + 110 + 150Adding these up:50 + 70 = 120120 + 100 = 220220 + 80 = 300300 + 120 = 420420 + 140 = 560560 + 160 = 720720 + 130 = 850850 + 110 = 960960 + 150 = 1110So, ( sum M_i = 1110 )Now, sum of ( E_i M_i ):I need to multiply each ( E_i ) by its corresponding ( M_i ) and then sum them all.Let's compute each product:1. 2 * 50 = 1002. 3 * 70 = 2103. 5 * 100 = 5004. 4 * 80 = 3205. 6 * 120 = 7206. 7 * 140 = 9807. 8 * 160 = 12808. 6 * 130 = 7809. 5 * 110 = 55010. 7 * 150 = 1050Now, adding these products together:100 + 210 = 310310 + 500 = 810810 + 320 = 11301130 + 720 = 18501850 + 980 = 28302830 + 1280 = 41104110 + 780 = 48904890 + 550 = 54405440 + 1050 = 6490So, ( sum E_i M_i = 6490 )Next, sum of ( E_i^2 ):Compute each ( E_i ) squared and then sum them.1. ( 2^2 = 4 )2. ( 3^2 = 9 )3. ( 5^2 = 25 )4. ( 4^2 = 16 )5. ( 6^2 = 36 )6. ( 7^2 = 49 )7. ( 8^2 = 64 )8. ( 6^2 = 36 )9. ( 5^2 = 25 )10. ( 7^2 = 49 )Adding these up:4 + 9 = 1313 + 25 = 3838 + 16 = 5454 + 36 = 9090 + 49 = 139139 + 64 = 203203 + 36 = 239239 + 25 = 264264 + 49 = 313So, ( sum E_i^2 = 313 )Now, plugging these into the formula for ( a ):[a = frac{nsum (E_i M_i) - sum E_i sum M_i}{nsum E_i^2 - (sum E_i)^2}]Given that ( n = 10 ), let's compute numerator and denominator separately.Numerator:( 10 * 6490 - 53 * 1110 )Compute 10 * 6490 = 64,900Compute 53 * 1110:First, 50 * 1110 = 55,500Then, 3 * 1110 = 3,330Total: 55,500 + 3,330 = 58,830So, numerator = 64,900 - 58,830 = 6,070Denominator:( 10 * 313 - (53)^2 )Compute 10 * 313 = 3,130Compute 53^2: 53 * 5350*50 = 2,50050*3 = 1503*50 = 1503*3 = 9So, 2,500 + 150 + 150 + 9 = 2,809Wait, that's not correct. Wait, 53^2 is actually 2,809. Let me verify:53 * 53:50*50 = 2,50050*3 = 1503*50 = 1503*3 = 9So, 2,500 + 150 + 150 + 9 = 2,809. Yes, that's correct.So, denominator = 3,130 - 2,809 = 321Therefore, ( a = frac{6,070}{321} )Let me compute that:Divide 6,070 by 321.321 * 19 = 6,099, which is a bit more than 6,070.So, 321 * 18 = 5,7786,070 - 5,778 = 292So, 18 + (292/321) ≈ 18 + 0.909 ≈ 18.909Wait, but 321 * 18.909 ≈ 6,070?Wait, maybe I should do a more precise division.Alternatively, perhaps I made a mistake in the numerator or denominator.Wait, let me double-check the numerator and denominator.Numerator: 10 * 6490 = 64,900Sum E_i = 53, sum M_i = 111053 * 1110: Let's compute 53 * 1000 = 53,000; 53 * 110 = 5,830; so total is 53,000 + 5,830 = 58,830So, 64,900 - 58,830 = 6,070. That seems correct.Denominator: 10 * 313 = 3,130Sum E_i squared is 313, so 10 * 313 = 3,130Sum E_i squared is 313, so (sum E_i)^2 is 53^2 = 2,809Thus, denominator is 3,130 - 2,809 = 321. Correct.So, a = 6,070 / 321Let me compute 321 * 18 = 5,7786,070 - 5,778 = 292So, 292 / 321 ≈ 0.9096So, a ≈ 18 + 0.9096 ≈ 18.9096So, approximately 18.91Wait, that seems high. Let me check the calculations again.Wait, 321 * 18.91 ≈ 321 * 18 + 321 * 0.91321 * 18 = 5,778321 * 0.91 ≈ 321 * 0.9 = 288.9; 321 * 0.01 = 3.21; so total ≈ 288.9 + 3.21 ≈ 292.11So, total ≈ 5,778 + 292.11 ≈ 6,070.11, which matches the numerator. So, a ≈ 18.91Wait, but 18.91 seems quite steep. Let me think about the data.Looking at the data, when E increases, M also increases. For example, from E=2 to E=3, M goes from 50 to 70, which is an increase of 20. From E=3 to E=5, M goes from 70 to 100, which is an increase of 30 over 2 events, so 15 per event. Then from E=5 to E=4, M decreases from 100 to 80, which is a decrease of 20 over -1 event, so 20 per event. Then from E=4 to E=6, M increases from 80 to 120, which is 40 over 2 events, 20 per event. Then E=6 to E=7, M from 120 to 140, 20 over 1 event. E=7 to E=8, M from 140 to 160, 20 over 1 event. E=8 to E=6, M from 160 to 130, decrease of 30 over -2 events, so 15 per event. E=6 to E=5, M from 130 to 110, decrease of 20 over -1 event, 20 per event. E=5 to E=7, M from 110 to 150, increase of 40 over 2 events, 20 per event.So, on average, the slope seems to be around 20 per event. So, 18.91 is close to that. Maybe that's correct.But let me double-check the calculations.Wait, 6,070 divided by 321.Let me compute 321 * 18 = 5,7786,070 - 5,778 = 292292 / 321 ≈ 0.9096So, 18.9096, which is approximately 18.91.So, a ≈ 18.91Now, compute ( b ):[b = frac{sum M_i - a sum E_i}{n}]We have ( sum M_i = 1110 ), ( sum E_i = 53 ), ( n = 10 ), and ( a ≈ 18.91 )Compute ( a * sum E_i ):18.91 * 53Compute 18 * 53 = 9540.91 * 53 ≈ 48.23So, total ≈ 954 + 48.23 ≈ 1,002.23So, ( sum M_i - a sum E_i ≈ 1110 - 1,002.23 ≈ 107.77 )Then, ( b ≈ 107.77 / 10 ≈ 10.777 )So, approximately 10.78Therefore, the regression line is:( M = 18.91 E + 10.78 )Wait, let me check if I did that correctly.Alternatively, maybe I should use more precise values.Wait, 6,070 / 321 is exactly:Let me perform the division more accurately.321 ) 6070321 * 18 = 5,7786070 - 5778 = 292Bring down a zero: 2920321 * 9 = 2,8892920 - 2889 = 31Bring down a zero: 310321 * 0.96 ≈ 308.16So, 321 * 0.96 ≈ 308.16So, 310 - 308.16 ≈ 1.84So, total is 18.96Wait, so 321 * 18.96 ≈ 6,070So, a ≈ 18.96Similarly, let's compute ( a * sum E_i ):18.96 * 53Compute 18 * 53 = 9540.96 * 53 = 51.68Total = 954 + 51.68 = 1,005.68Then, ( sum M_i - a sum E_i = 1110 - 1,005.68 = 104.32 )Then, ( b = 104.32 / 10 = 10.432 )So, approximately 10.43Therefore, the regression line is:( M = 18.96 E + 10.43 )Rounding to two decimal places, perhaps 18.96 and 10.43.But let me check if I can represent this more accurately.Alternatively, maybe I should keep more decimal places during calculations to avoid rounding errors.Alternatively, perhaps I should use fractions.But maybe for simplicity, let's keep it to two decimal places.So, a ≈ 18.96, b ≈ 10.43Therefore, the regression line is approximately ( M = 18.96 E + 10.43 )But let me check if this makes sense.Looking at the data, when E=2, M=50.Plugging into the equation: 18.96*2 + 10.43 = 37.92 + 10.43 ≈ 48.35, which is close to 50.When E=7, M=140.18.96*7 ≈ 132.72 + 10.43 ≈ 143.15, which is close to 140.Similarly, E=8, M=160.18.96*8 ≈ 151.68 + 10.43 ≈ 162.11, which is close to 160.So, seems reasonable.Alternatively, maybe I should compute using more precise values.But perhaps I can use the exact fractions.Wait, 6,070 / 321.Let me see if 321 divides 6,070 evenly.321 * 18 = 5,7786,070 - 5,778 = 292So, 292 / 321 = 292/321Simplify: divide numerator and denominator by GCD(292,321). Let's see, 321 - 292 = 29. 292 ÷ 29 = 10.06... Not integer. 29 divides 292? 29*10=290, so 292-290=2, so no. So, 292 and 321 have GCD 1. So, 292/321 is the simplest form.So, a = 18 + 292/321 ≈ 18.9096Similarly, b = (1110 - a*53)/10Compute a*53:18*53 = 954(292/321)*53 = (292*53)/321Compute 292*53:292*50=14,600292*3=876Total=14,600 + 876=15,476So, (15,476)/321 ≈ 48.2118So, a*53 ≈ 954 + 48.2118 ≈ 1,002.2118Then, 1110 - 1,002.2118 ≈ 107.7882Divide by 10: 10.77882So, b ≈ 10.7788So, more accurately, a ≈ 18.9096 and b ≈ 10.7788So, the regression line is:( M = 18.9096 E + 10.7788 )Rounding to four decimal places, perhaps 18.9096 and 10.7788.But for simplicity, maybe we can round to two decimal places: 18.91 and 10.78.So, ( M = 18.91 E + 10.78 )Alternatively, if we want to keep more decimals, but perhaps two is sufficient.Now, moving to part 2: Maria wants to understand the effect of doubling the number of outreach events on community engagement. Using the regression line, calculate the expected increase in M if E is doubled from the average number of events per month.First, I need to find the average number of events per month, which is ( bar{E} = sum E_i / n = 53 / 10 = 5.3 )So, the average E is 5.3.Doubling that would be 10.6.But wait, in the context of the problem, E is the number of events, which are integers. But since we're using a regression model, we can use non-integer values.So, the current average is 5.3, doubling it would be 10.6.But wait, let me think. Is the question asking for doubling the average, or doubling the number of events from the average? I think it's the latter: doubling the number of events from the average.So, if the average is 5.3, then doubling it would be 5.3 * 2 = 10.6.But let me check: "the number of outreach events (E) is doubled from the average number of events held per month over the 10-month period."Yes, so E is doubled from the average, so E_new = 2 * average_E = 2 * 5.3 = 10.6Now, using the regression line, compute M at E=10.6 and at E=5.3, then find the difference.Alternatively, since the regression line is linear, the change in M when E increases by ΔE is a * ΔE.So, if E increases from 5.3 to 10.6, ΔE = 5.3, so the expected increase in M is a * 5.3.Given that a ≈ 18.91, the increase would be 18.91 * 5.3 ≈ ?Compute 18.91 * 5 = 94.5518.91 * 0.3 = 5.673Total ≈ 94.55 + 5.673 ≈ 100.223So, approximately 100.22 increase in M.Alternatively, compute M at E=10.6 and E=5.3.Compute M at E=5.3:M = 18.91 * 5.3 + 10.78Compute 18.91 * 5 = 94.5518.91 * 0.3 = 5.673Total: 94.55 + 5.673 = 100.223Add 10.78: 100.223 + 10.78 ≈ 110.003So, M ≈ 110.003 when E=5.3Wait, but that's interesting because the average M is 1110 / 10 = 111. So, the regression line predicts M ≈ 110 when E=5.3, which is close to the actual average M of 111. That makes sense because the regression line passes through the point (average E, average M).Now, compute M at E=10.6:M = 18.91 * 10.6 + 10.78Compute 18.91 * 10 = 189.118.91 * 0.6 = 11.346Total: 189.1 + 11.346 = 200.446Add 10.78: 200.446 + 10.78 ≈ 211.226So, M ≈ 211.226 when E=10.6The increase is 211.226 - 110.003 ≈ 101.223Wait, that's slightly more than the previous calculation because of rounding errors. But essentially, the increase is approximately 101.22.So, the expected increase in M is approximately 101.22.But let me compute it more accurately using the exact a and b values.We had a ≈ 18.9096 and b ≈ 10.7788Compute M at E=5.3:M = 18.9096 * 5.3 + 10.7788Compute 18.9096 * 5 = 94.54818.9096 * 0.3 = 5.67288Total: 94.548 + 5.67288 = 100.22088Add 10.7788: 100.22088 + 10.7788 ≈ 111.0Which is exactly the average M, as expected.Now, M at E=10.6:M = 18.9096 * 10.6 + 10.7788Compute 18.9096 * 10 = 189.09618.9096 * 0.6 = 11.34576Total: 189.096 + 11.34576 = 200.44176Add 10.7788: 200.44176 + 10.7788 ≈ 211.22056So, M ≈ 211.22056The increase is 211.22056 - 111.0 ≈ 100.22056So, approximately 100.22 increase.Therefore, the expected increase in M is approximately 100.22.So, rounding to two decimal places, 100.22.Alternatively, since the question might expect an integer, perhaps 100 or 100.22.But let me check if I did everything correctly.Wait, another approach: since the regression line is linear, the change in M when E increases by ΔE is a * ΔE.Here, ΔE = 10.6 - 5.3 = 5.3So, change in M = a * 5.3 = 18.9096 * 5.3Compute 18.9096 * 5 = 94.54818.9096 * 0.3 = 5.67288Total: 94.548 + 5.67288 = 100.22088So, same result.Therefore, the expected increase is approximately 100.22.So, summarizing:1. The regression line is ( M = 18.91 E + 10.78 )2. The expected increase in M when doubling E from the average is approximately 100.22.But let me present the answers with more precise decimals if possible.Alternatively, perhaps we can keep more decimals in a and b.But for the sake of the answer, let's present them as:1. ( M = 18.91 E + 10.78 )2. The expected increase is approximately 100.22.Alternatively, if we want to present exact fractions, but that might be complicated.Alternatively, perhaps we can compute a and b more precisely.But given the time, I think two decimal places are sufficient.So, final answers:1. The regression line is ( M = 18.91 E + 10.78 )2. The expected increase is approximately 100.22 community members engaged.But let me check if the question expects the answer in a specific format.Wait, the question says "calculate the expected increase in the number of community members engaged (M) if the number of outreach events (E) is doubled from the average number of events held per month over the 10-month period."So, the increase is 100.22, which we can round to 100.22 or perhaps 100.2.Alternatively, if we use the exact a value:a = 6,070 / 321 ≈ 18.90965732So, a ≈ 18.90965732Then, ΔE = 5.3So, ΔM = a * ΔE = 18.90965732 * 5.3Compute 18.90965732 * 5 = 94.548286618.90965732 * 0.3 = 5.6728972Total: 94.5482866 + 5.6728972 ≈ 100.2211838So, approximately 100.2212So, rounding to two decimal places, 100.22.Alternatively, if we round to the nearest whole number, 100.But since the original data has M as whole numbers, but the regression can predict non-integers, so 100.22 is acceptable.Therefore, the answers are:1. The regression line is ( M = 18.91 E + 10.78 )2. The expected increase is approximately 100.22 community members engaged.</think>