🧠:<think>Okay, so I have this problem where a departmental manager is trying to develop a new product. The success is measured by two things: time to market (T) and product efficiency (E). The manager can hire up to 3 additional developers (x) and allocate up to 10 overtime hours per week (y). Both x and y are non-negative integers. The first part is to find the optimal x and y that minimize T(x, y) = 15 + 4x² - y, while making sure that E(x, y) = 3xy - 2x² + 5y is at least 50. The second part is to find the minimum cost using the cost function C(x, y) = 2000x + 150y.Alright, let's start with the first part. I need to minimize T(x, y) while ensuring E(x, y) ≥ 50. Since x and y are integers between 0 and 3 for x, and 0 to 10 for y, I can probably approach this by enumerating possible values of x and y within these ranges, compute T and E for each, and then find the combination where T is minimized but E is at least 50.But before jumping into enumeration, maybe I can analyze the functions a bit to narrow down the possibilities.Looking at T(x, y) = 15 + 4x² - y. To minimize T, we want to minimize 4x² and maximize y. So, ideally, we'd want x to be as small as possible and y as large as possible. However, E(x, y) must be at least 50, so we can't just set x=0 and y=10 because E might not meet the requirement.Looking at E(x, y) = 3xy - 2x² + 5y. Let's see how E behaves with different x and y. For a given x, E increases with y because of the 3xy and 5y terms. So for each x, the higher y is, the higher E will be. Therefore, for each x, the minimum y needed to get E ≥ 50 can be found, and then we can compute T for that x and y, and choose the x that gives the smallest T.So, perhaps the strategy is:1. For each x from 0 to 3: a. Find the smallest y such that E(x, y) ≥ 50. b. If such a y exists within the 0 to 10 range, compute T(x, y). c. If no such y exists, then that x is not feasible.2. Among all feasible (x, y) pairs, choose the one with the smallest T(x, y).Let me try that.Starting with x=0:E(0, y) = 3*0*y - 2*0² + 5y = 5y. We need 5y ≥ 50, so y ≥ 10. Since y can be at most 10, y=10 is the only possibility.Compute T(0,10) = 15 + 4*0² - 10 = 15 -10 = 5 days.Okay, that's a candidate.Now x=1:E(1, y) = 3*1*y - 2*1² + 5y = 3y - 2 + 5y = 8y - 2. We need 8y - 2 ≥ 50 => 8y ≥ 52 => y ≥ 6.5. Since y must be integer, y ≥7.So y can be 7,8,9,10.Compute T(1,7) = 15 + 4*1 -7 = 15 +4 -7=12T(1,8)=15+4 -8=11T(1,9)=15+4 -9=10T(1,10)=15+4 -10=9So the minimum T for x=1 is 9 when y=10.Compare with x=0: T=5. So x=0 is better.x=2:E(2, y)=3*2*y -2*4 +5y=6y -8 +5y=11y -8. We need 11y -8 ≥50 =>11y ≥58 => y ≥58/11≈5.27. So y≥6.Compute T(2,6)=15 + 4*4 -6=15+16-6=25T(2,7)=15+16 -7=24T(2,8)=15+16 -8=23T(2,9)=15+16 -9=22T(2,10)=15+16 -10=21So the minimum T for x=2 is 21 when y=10.Compare with x=0: T=5 is still better.x=3:E(3, y)=3*3*y -2*9 +5y=9y -18 +5y=14y -18. We need 14y -18 ≥50 =>14y ≥68 => y≥68/14≈4.857. So y≥5.Compute T(3,5)=15 +4*9 -5=15+36 -5=46T(3,6)=15+36 -6=45T(3,7)=15+36 -7=44T(3,8)=15+36 -8=43T(3,9)=15+36 -9=42T(3,10)=15+36 -10=41So the minimum T for x=3 is 41 when y=10.So summarizing:x=0: T=5, y=10x=1: T=9, y=10x=2: T=21, y=10x=3: T=41, y=10So clearly, x=0 and y=10 gives the minimum T=5.But wait, is E(0,10)=50? Let me check:E(0,10)=5*10=50. Exactly 50, which meets the requirement.So, that seems to be the optimal solution.Wait, but before finalizing, let me check if for x=0, y=10 is the only possible y. Since E(0,y)=5y, so y must be at least 10 to get E=50. Since y cannot exceed 10, y=10 is the only option.So, the optimal values are x=0, y=10.But wait, let me think again. Is there a possibility that for some x>0, with a lower y, we can get E≥50 and have a lower T?Wait, for x=1, y=7 gives E=8*7 -2=56-2=54, which is above 50, and T=12. But 12 is higher than 5, so x=0 is better.Similarly, for x=2, y=6 gives E=11*6 -8=66-8=58, which is above 50, and T=25, which is higher than 5.Same for x=3, y=5 gives E=14*5 -18=70-18=52, which is above 50, and T=46, which is way higher.So, indeed, x=0, y=10 is the optimal.Wait, but let me check if for x=0, y=10 is the only solution. What if x=0, y=10 is the only way to get E=50? Yes, because E(0,y)=5y, so y must be 10.Alternatively, is there a way to have x>0 and y<10 but still get E≥50? For example, x=1, y=7 gives E=54, which is above 50, but T=12, which is worse than x=0, y=10.Similarly, x=2, y=6 gives E=58, T=25.So, even though for x>0, we can have lower y and still meet E≥50, the T(x,y) increases, so x=0, y=10 is still better.Therefore, the optimal solution is x=0, y=10.Now, moving to part 2: compute the minimum cost C(x,y)=2000x +150y.Given x=0, y=10:C=2000*0 +150*10=0 +1500=1500 dollars.Is that the minimum cost? Well, since x=0 and y=10 is the optimal for the first part, yes, that's the cost.Wait, but let me think again. Is there a way to have a lower cost? For example, if we can have a lower x and y that still meets E≥50, but with lower cost.But in this case, x=0, y=10 is the only way to get E=50, and any other combination with x>0 would require higher y or higher x, both of which would increase the cost.Wait, actually, for x=1, y=7, E=54, which is above 50, and cost would be 2000*1 +150*7=2000 +1050=3050, which is higher than 1500.Similarly, x=2, y=6: cost=4000 +900=4900.x=3, y=5: cost=6000 +750=6750.So, indeed, x=0, y=10 gives the minimum cost of 1500.Therefore, the answers are x=0, y=10, and cost=1500.But wait, let me double-check if x=0, y=10 is indeed feasible. Yes, x can be 0 (since it's non-negative), and y=10 is within the maximum allowed.Also, E(0,10)=50, which meets the requirement.So, I think that's solid.Final Answer1. The optimal values are ( x = boxed{0} ) and ( y = boxed{10} ).2. The minimum cost is ( boxed{1500} ) dollars.</think>

🧠:<think>Alright, so I have this problem about Alex, a military spouse who's moved around a lot because of their partner's assignments. Each time they move, Alex has to find a new healthcare provider, and the time it takes varies depending on the community. The problem is divided into two parts. Let me try to tackle them one by one.Starting with part 1: I need to find the total average time (T_{text{total}}) it took for Alex to find healthcare providers across all (n) communities. The average time in each community (i) is given by the function (T_i = frac{50}{sqrt{i}} + frac{100}{i}). So, I think I need to sum this function from (i = 1) to (i = n).So, mathematically, (T_{text{total}} = sum_{i=1}^{n} T_i = sum_{i=1}^{n} left( frac{50}{sqrt{i}} + frac{100}{i} right)). That simplifies to (T_{text{total}} = 50 sum_{i=1}^{n} frac{1}{sqrt{i}} + 100 sum_{i=1}^{n} frac{1}{i}).Hmm, okay. So, I need to express this as a summation and simplify as much as possible. I remember that the sum of (1/sqrt{i}) from 1 to n is a known series, but I don't recall the exact formula. Similarly, the sum of (1/i) is the harmonic series, which also doesn't have a simple closed-form expression, but it can be approximated using the natural logarithm.Let me think. The harmonic series (sum_{i=1}^{n} frac{1}{i}) is approximately (ln(n) + gamma), where (gamma) is the Euler-Mascheroni constant, approximately 0.5772. But since the problem says to express it as a summation and simplify as much as possible, maybe they just want me to write it in terms of known series or approximate expressions.Similarly, the sum (sum_{i=1}^{n} frac{1}{sqrt{i}}) can be approximated by an integral. I remember that for decreasing functions, the sum can be approximated by the integral from 1 to n of (1/sqrt{x}) dx, which is (2sqrt{n} - 2). But again, since it's a sum, maybe they just want the expression in summation form or perhaps an approximation.Wait, the problem says "express (T_{text{total}}) as a summation and simplify as much as possible." So, maybe I don't need to approximate it but just express it in terms of summations. Let me check.So, (T_{text{total}} = 50 sum_{i=1}^{n} frac{1}{sqrt{i}} + 100 sum_{i=1}^{n} frac{1}{i}). That's the summation expression. Is there a way to simplify this further? I don't think so because both sums don't have simple closed-form expressions. So, perhaps this is the simplified form as much as possible.But wait, maybe I can factor out something. Let me see. Both terms have a factor of 50 and 100, which are constants. So, I can write it as (50 sum_{i=1}^{n} frac{1}{sqrt{i}} + 100 sum_{i=1}^{n} frac{1}{i}). That seems as simplified as it can get without approximating.Alternatively, I can factor out 50 from both terms: (50 left( sum_{i=1}^{n} frac{1}{sqrt{i}} + 2 sum_{i=1}^{n} frac{1}{i} right)). Hmm, that might be a slightly more compact form, but I'm not sure if that's necessary. The problem just says to express it as a summation and simplify as much as possible, so either form is acceptable, I think.Moving on to part 2: Now, Alex's relocations follow a Poisson distribution with an average rate of (lambda) relocations per year. The probability of having relocated to exactly (k) communities in the past 10 years is given by (P(k) = frac{e^{-lambda cdot 10} (lambda cdot 10)^k}{k!}). I need to find the expected total average time (E[T_{text{total}}]) for Alex to find healthcare providers over the past 10 years.Okay, so (T_{text{total}}) is a random variable that depends on (k), the number of relocations. Since (k) follows a Poisson distribution with parameter (mu = lambda cdot 10), the expected value (E[T_{text{total}}]) is the sum over all possible (k) of (T_{text{total}}(k) cdot P(k)).From part 1, (T_{text{total}}(k) = 50 sum_{i=1}^{k} frac{1}{sqrt{i}} + 100 sum_{i=1}^{k} frac{1}{i}). So, the expectation is:(E[T_{text{total}}] = sum_{k=0}^{infty} left( 50 sum_{i=1}^{k} frac{1}{sqrt{i}} + 100 sum_{i=1}^{k} frac{1}{i} right) cdot frac{e^{-10lambda} (10lambda)^k}{k!}).Hmm, that looks complicated. Maybe I can interchange the order of summation or find a way to express this expectation in terms of known quantities.Let me think about linearity of expectation. Since expectation is linear, I can write:(E[T_{text{total}}] = 50 Eleft[ sum_{i=1}^{k} frac{1}{sqrt{i}} right] + 100 Eleft[ sum_{i=1}^{k} frac{1}{i} right]).Which is equal to:(50 sum_{i=1}^{infty} frac{1}{sqrt{i}} E[I_{{k geq i}}] + 100 sum_{i=1}^{infty} frac{1}{i} E[I_{{k geq i}}]).Wait, that might not be the right approach. Alternatively, since (k) is a Poisson random variable, perhaps I can find the expectation by recognizing that for each (i), the term (frac{1}{sqrt{i}}) is added if (k geq i). So, the expectation can be written as:(E[T_{text{total}}] = 50 sum_{i=1}^{infty} frac{1}{sqrt{i}} P(k geq i) + 100 sum_{i=1}^{infty} frac{1}{i} P(k geq i)).But (P(k geq i)) is the probability that the number of relocations is at least (i). For a Poisson distribution, (P(k geq i) = 1 - P(k leq i - 1)). However, calculating this for each (i) might not lead to a simple expression.Alternatively, maybe we can express the expectation as:(E[T_{text{total}}] = 50 sum_{i=1}^{infty} frac{1}{sqrt{i}} E[I_{{k geq i}}] + 100 sum_{i=1}^{infty} frac{1}{i} E[I_{{k geq i}}]).Where (I_{{k geq i}}) is an indicator variable that is 1 if (k geq i) and 0 otherwise. Then, (E[I_{{k geq i}}] = P(k geq i)).But again, without a closed-form expression for (P(k geq i)), this might not help much. Maybe there's another approach.Wait, perhaps we can use the fact that for a Poisson random variable (k) with parameter (mu = 10lambda), the expectation (E[g(k)]) can sometimes be expressed in terms of generating functions or other properties.Alternatively, maybe we can switch the order of summation. Let me try that.Starting from:(E[T_{text{total}}] = sum_{k=0}^{infty} left( 50 sum_{i=1}^{k} frac{1}{sqrt{i}} + 100 sum_{i=1}^{k} frac{1}{i} right) cdot frac{e^{-10lambda} (10lambda)^k}{k!}).Let me separate the two terms:(E[T_{text{total}}] = 50 sum_{k=0}^{infty} sum_{i=1}^{k} frac{1}{sqrt{i}} cdot frac{e^{-10lambda} (10lambda)^k}{k!} + 100 sum_{k=0}^{infty} sum_{i=1}^{k} frac{1}{i} cdot frac{e^{-10lambda} (10lambda)^k}{k!}).Now, I can switch the order of summation. For the first term, the inner sum is over (i) from 1 to (k), so if I switch, (i) will go from 1 to (infty), and for each (i), (k) goes from (i) to (infty). So:First term becomes:(50 sum_{i=1}^{infty} frac{1}{sqrt{i}} sum_{k=i}^{infty} frac{e^{-10lambda} (10lambda)^k}{k!}).Similarly, the second term becomes:(100 sum_{i=1}^{infty} frac{1}{i} sum_{k=i}^{infty} frac{e^{-10lambda} (10lambda)^k}{k!}).Now, the inner sum (sum_{k=i}^{infty} frac{e^{-10lambda} (10lambda)^k}{k!}) is equal to (P(k geq i)), which is the survival function of the Poisson distribution. But I don't think there's a simple closed-form for this.However, I recall that for a Poisson distribution with parameter (mu), the expectation (E[I_{{k geq i}}]) is equal to (1 - sum_{j=0}^{i-1} frac{e^{-mu} mu^j}{j!}). But that still doesn't give me a simple expression.Wait, maybe there's another way. Let me think about the generating function of the Poisson distribution. The probability generating function is (G(t) = e^{mu(t - 1)}). But I'm not sure if that helps here.Alternatively, maybe I can express the inner sum in terms of the incomplete gamma function. Because the sum (sum_{k=i}^{infty} frac{mu^k}{k!}) is related to the upper incomplete gamma function. Specifically, (sum_{k=i}^{infty} frac{mu^k}{k!} = frac{Gamma(i, mu)}{(i-1)!}), where (Gamma(i, mu)) is the upper incomplete gamma function.But I'm not sure if that's helpful in this context because it's still a special function and not a simple expression. Maybe the problem expects an answer in terms of these summations or perhaps an approximate expression.Alternatively, maybe I can use the fact that for a Poisson distribution, the expectation (E[f(k)]) can sometimes be expressed as a sum involving (f(k)) multiplied by the Poisson probability mass function. But in this case, (f(k)) is itself a sum, so it's a double summation.Wait, perhaps I can write the expectation as:(E[T_{text{total}}] = 50 Eleft[ sum_{i=1}^{k} frac{1}{sqrt{i}} right] + 100 Eleft[ sum_{i=1}^{k} frac{1}{i} right]).Which is equal to:(50 sum_{i=1}^{infty} frac{1}{sqrt{i}} P(k geq i) + 100 sum_{i=1}^{infty} frac{1}{i} P(k geq i)).But again, without knowing (P(k geq i)) in a closed form, this might not help. Alternatively, perhaps I can use the fact that for a Poisson distribution, (E[k] = mu = 10lambda), but that's just the mean. However, the expectation of a sum up to (k) is not straightforward.Wait, maybe I can think of it as:(Eleft[ sum_{i=1}^{k} frac{1}{sqrt{i}} right] = sum_{i=1}^{infty} frac{1}{sqrt{i}} P(k geq i)).Similarly for the harmonic series. So, perhaps I can write:(E[T_{text{total}}] = 50 sum_{i=1}^{infty} frac{1}{sqrt{i}} P(k geq i) + 100 sum_{i=1}^{infty} frac{1}{i} P(k geq i)).But I still need to find (P(k geq i)) for each (i). Since (k) is Poisson with (mu = 10lambda), (P(k geq i) = 1 - sum_{j=0}^{i-1} frac{e^{-10lambda} (10lambda)^j}{j!}).So, substituting that in, we get:(E[T_{text{total}}] = 50 sum_{i=1}^{infty} frac{1}{sqrt{i}} left(1 - sum_{j=0}^{i-1} frac{e^{-10lambda} (10lambda)^j}{j!}right) + 100 sum_{i=1}^{infty} frac{1}{i} left(1 - sum_{j=0}^{i-1} frac{e^{-10lambda} (10lambda)^j}{j!}right)).This seems quite complicated, but maybe it's the most precise expression we can get without further simplification. Alternatively, perhaps we can approximate these sums using integrals or known approximations for the Poisson distribution.Wait, another thought: since (k) is Poisson with mean (10lambda), for large (lambda), the distribution can be approximated by a normal distribution with mean (10lambda) and variance (10lambda). But I'm not sure if that's helpful here because we're dealing with the expectation of sums up to (k), which might not have a straightforward approximation.Alternatively, maybe we can use the fact that for a Poisson random variable (k), the expectation (E[f(k)]) can sometimes be expressed in terms of the derivative of the generating function. But I'm not sure how to apply that here.Wait, let me think differently. Maybe I can express the expectation as:(E[T_{text{total}}] = 50 Eleft[ sum_{i=1}^{k} frac{1}{sqrt{i}} right] + 100 Eleft[ sum_{i=1}^{k} frac{1}{i} right]).Which can be written as:(50 sum_{i=1}^{infty} frac{1}{sqrt{i}} P(k geq i) + 100 sum_{i=1}^{infty} frac{1}{i} P(k geq i)).But I still need to compute these sums. Alternatively, maybe I can recognize that (Eleft[ sum_{i=1}^{k} frac{1}{sqrt{i}} right] = sum_{i=1}^{infty} frac{1}{sqrt{i}} P(k geq i)), which is the same as (sum_{i=1}^{infty} frac{1}{sqrt{i}} (1 - F_{k}(i-1))), where (F_{k}(i-1)) is the CDF of (k) evaluated at (i-1).But without knowing a closed-form for (F_{k}(i-1)), this might not help. Alternatively, perhaps we can use generating functions or other techniques, but I'm not sure.Wait, maybe I can use the fact that for a Poisson distribution, the probability (P(k geq i)) can be expressed as (1 - sum_{j=0}^{i-1} frac{e^{-mu} mu^j}{j!}), where (mu = 10lambda). So, substituting that in, we get:(E[T_{text{total}}] = 50 sum_{i=1}^{infty} frac{1}{sqrt{i}} left(1 - sum_{j=0}^{i-1} frac{e^{-10lambda} (10lambda)^j}{j!}right) + 100 sum_{i=1}^{infty} frac{1}{i} left(1 - sum_{j=0}^{i-1} frac{e^{-10lambda} (10lambda)^j}{j!}right)).This is a precise expression, but it's quite involved. I don't think it can be simplified further without approximations or special functions.Alternatively, maybe I can approximate the sums for large (n). For example, if (10lambda) is large, the Poisson distribution can be approximated by a normal distribution, and the sums can be approximated by integrals. But I'm not sure if that's within the scope of this problem.Wait, another idea: perhaps I can express the expectation in terms of the expectation of (k). Since (E[k] = 10lambda), maybe I can find an expression involving (10lambda). But I don't see an immediate way to connect (E[T_{text{total}}]) to (E[k]) because (T_{text{total}}) is a sum up to (k), not a function of (k) itself.Hmm, maybe I'm overcomplicating this. Let me go back to the original expression:(E[T_{text{total}}] = sum_{k=0}^{infty} left( 50 sum_{i=1}^{k} frac{1}{sqrt{i}} + 100 sum_{i=1}^{k} frac{1}{i} right) cdot frac{e^{-10lambda} (10lambda)^k}{k!}).Perhaps I can factor out the constants:(E[T_{text{total}}] = 50 sum_{k=0}^{infty} sum_{i=1}^{k} frac{1}{sqrt{i}} cdot frac{e^{-10lambda} (10lambda)^k}{k!} + 100 sum_{k=0}^{infty} sum_{i=1}^{k} frac{1}{i} cdot frac{e^{-10lambda} (10lambda)^k}{k!}).Now, switching the order of summation as I did before:(E[T_{text{total}}] = 50 sum_{i=1}^{infty} frac{1}{sqrt{i}} sum_{k=i}^{infty} frac{e^{-10lambda} (10lambda)^k}{k!} + 100 sum_{i=1}^{infty} frac{1}{i} sum_{k=i}^{infty} frac{e^{-10lambda} (10lambda)^k}{k!}).So, each term is now a sum over (i) multiplied by the survival function of the Poisson distribution at (i). I think this is as far as I can go without more advanced techniques or approximations.Alternatively, maybe I can recognize that the inner sum (sum_{k=i}^{infty} frac{e^{-10lambda} (10lambda)^k}{k!}) is equal to (P(k geq i)), which is the same as (1 - P(k leq i - 1)). But since I can't express this in a simpler form, I think this is the most precise answer I can give.So, putting it all together, the expected total average time is:(E[T_{text{total}}] = 50 sum_{i=1}^{infty} frac{1}{sqrt{i}} P(k geq i) + 100 sum_{i=1}^{infty} frac{1}{i} P(k geq i)).Where (P(k geq i) = 1 - sum_{j=0}^{i-1} frac{e^{-10lambda} (10lambda)^j}{j!}).I think this is the most simplified form I can get for the expectation without further approximations or special functions.</think>

🧠:<think>Okay, so I'm trying to figure out how extreme couponing affects the revenue of a supermarket. The manager wants to evaluate this impact, and there are two parts to the problem. Let me take them one at a time.Problem 1: Calculating Total Revenue from a Customer Using Maximum CouponsAlright, the first part says that the store allows a maximum of 3 coupons per item, and each coupon can give up to 25% off the item's base price. A customer buys 10 items, each priced at p, and uses the maximum coupons to get the highest possible discount per item. I need to express the total revenue R from this customer in terms of p.Hmm, so each item can have up to 3 coupons. Each coupon gives 25% off, which is a discount of 0.25p per coupon. But wait, if you apply multiple coupons, does each subsequent coupon apply to the discounted price or the original price? That's an important detail because it affects the total discount.I think in most cases, coupons stack, but sometimes they might be applied to the original price or the discounted price. Let me clarify. If each coupon is 25% off, and you can use 3 coupons per item, then each coupon would reduce the price by 25%. So, the first coupon takes 25% off, the second another 25%, and the third another 25%.But wait, if you apply 25% off three times, does that mean the total discount is 75%? Because 25% + 25% + 25% = 75%. So, each item would be discounted by 75%, meaning the customer pays 25% of the original price.Let me double-check that. If the original price is p, applying one coupon gives p - 0.25p = 0.75p. Applying a second coupon would take 25% off the new price, which is 0.75p * 0.75 = 0.5625p. Applying a third coupon would take another 25%, so 0.5625p * 0.75 = 0.421875p.Wait, so actually, each subsequent coupon is applied to the already discounted price, which means the total discount isn't just 75%, but it's compounded. So, the final price after three coupons is p * (0.75)^3.Calculating that: 0.75^3 = 0.421875. So, each item is priced at 0.421875p after three coupons.Therefore, for each item, the revenue is 0.421875p. Since the customer buys 10 items, the total revenue R would be 10 * 0.421875p.Let me compute that: 10 * 0.421875 = 4.21875. So, R = 4.21875p.But wait, maybe I should express it as a fraction instead of a decimal to make it cleaner. Let's see, 0.75 is 3/4, so (3/4)^3 is 27/64. So, each item is 27/64 of p. Therefore, 10 items would be 10*(27/64)p = 270/64 p. Simplifying that, 270 divided by 64 is 4.21875, which matches the decimal I had earlier.So, R = (270/64)p. Alternatively, I can write it as (135/32)p, since both numerator and denominator can be divided by 2. 270 ÷ 2 = 135, 64 ÷ 2 = 32. So, R = (135/32)p.Alternatively, as a decimal, it's approximately 4.21875p, but since the question asks for an expression in terms of p, either form is acceptable, but fractions are usually preferred in mathematical expressions.Wait, let me make sure I didn't make a mistake in the calculation. So, each coupon is 25%, so each time, the price is multiplied by 0.75. So, three coupons would be 0.75^3 = 0.421875. So, per item, it's 0.421875p, which is 27/64 p. So, 10 items would be 10*(27/64)p = 270/64 p = 135/32 p.Yes, that seems correct. So, R = (135/32)p.But wait, another thought: is the maximum discount per item 3 coupons, each of 25%, so 75% total discount? Or is it that each coupon can be up to 25%, but you can have multiple coupons, each potentially giving a different discount? But the problem says each coupon can provide a discount of up to 25%, so I think each coupon is 25%, so three coupons would be 75% discount.But wait, if each coupon is 25%, then the first coupon takes 25%, the second takes 25% of the original price or the discounted price? That's the key. If it's 25% off the original price each time, then three coupons would give 75% off the original price. But if each subsequent coupon is applied to the discounted price, then it's 25% off the already discounted price, leading to a total discount of 1 - (0.75)^3 = 1 - 0.421875 = 0.578125, which is about 57.8125%.So, which is it? The problem says each coupon can provide a discount of up to 25% off the item's base price. Hmm, the wording says "off the item's base price," which suggests that each coupon is 25% off the original price, not the discounted price.Wait, that would be different. So, if each coupon is 25% off the base price, then three coupons would give 3*25% = 75% off the base price, so the customer pays 25% of the original price.So, in that case, each item would be p - 3*(0.25p) = p - 0.75p = 0.25p. Therefore, each item is 0.25p, and 10 items would be 10*0.25p = 2.5p.Wait, that's a big difference. So, which interpretation is correct?The problem says: "each coupon can provide a discount of up to 25% off the item's base price." So, each coupon is 25% off the base price, not the current price. So, each coupon is 0.25p, regardless of previous discounts.Therefore, three coupons would give 3*0.25p = 0.75p discount, so the price paid is p - 0.75p = 0.25p per item.Therefore, for 10 items, the total revenue is 10*0.25p = 2.5p.Wait, that seems more straightforward. So, perhaps my initial thought about compounding was incorrect because the problem specifies that each coupon is off the base price, not the discounted price.So, if each coupon is 25% off the base price, then each coupon is 0.25p, so three coupons would be 0.75p, so the customer pays 0.25p per item.Therefore, 10 items would be 10*0.25p = 2.5p.So, R = 2.5p.But let me check the wording again: "each coupon can provide a discount of up to 25% off the item's base price." So, yes, each coupon is 25% off the base price, so they are additive, not multiplicative.Therefore, the total discount per item is 3*0.25p = 0.75p, so the price paid is p - 0.75p = 0.25p.Therefore, for 10 items, R = 10*0.25p = 2.5p.So, I think that's the correct approach. Initially, I thought it was compounding, but the wording suggests it's additive.So, the answer to part 1 is R = (5/2)p, since 2.5 is 5/2.Wait, 2.5 is 5/2? No, 5/2 is 2.5, yes. So, R = (5/2)p.Alternatively, R = 2.5p.But in fraction form, 5/2 is better.So, R = (5/2)p.Wait, but let me confirm once more. If each coupon is 25% off the base price, then three coupons would give 75% off the base price, so the customer pays 25% of the base price per item. So, 10 items would be 10*(0.25p) = 2.5p, which is 5/2 p.Yes, that seems correct.Problem 2: Calculating Total Loss in Revenue Due to Extreme Couponing Over a WeekNow, moving on to the second part. Over a week, the store has 1000 transactions, and 10% of these are from extreme couponers. So, 10% of 1000 is 100 transactions.Each of these extreme couponer transactions has an average base price of items purchased as p, and the average number of items per transaction is 15.We need to determine the total loss in revenue L for the store due to extreme couponing, in terms of p.Assuming that non-extreme couponers pay the full price, so their revenue is not affected. Only the extreme couponers cause the loss.First, let's figure out the revenue loss per transaction from an extreme couponer.From part 1, we know that each extreme couponer uses maximum coupons, which gives them a 75% discount per item, so they pay 25% of the base price per item.Therefore, for each item, the store loses 75% of p, which is 0.75p.But wait, in part 1, the customer bought 10 items, each at 0.25p, so the loss per item is 0.75p, and total loss per transaction was 10*0.75p = 7.5p.But in this case, the average number of items per transaction is 15, not 10. So, let's adjust accordingly.So, for each extreme couponer transaction, the store loses 0.75p per item, and with 15 items per transaction, the total loss per transaction is 15*0.75p = 11.25p.Therefore, each extreme couponer transaction causes a loss of 11.25p.Since there are 100 such transactions in a week, the total loss L is 100*11.25p = 1125p.But let me make sure I'm calculating this correctly.Alternatively, we can calculate the expected revenue without coupons and the actual revenue with coupons, then find the difference.Without coupons, each extreme couponer transaction would generate revenue of 15p (since each item is p, 15 items, no discount). But with coupons, they pay 0.25p per item, so 15*0.25p = 3.75p.Therefore, the loss per transaction is 15p - 3.75p = 11.25p, which matches the previous calculation.So, for 100 transactions, the total loss is 100*11.25p = 1125p.Therefore, L = 1125p.Wait, but let me think again. The problem says "the average base price of items purchased in these transactions is p." So, does that mean that each item's base price is p, or the average across all items is p? I think it's the latter, meaning that each item's base price is p, so the average is p.Therefore, each item is p, so the calculations hold.Alternatively, if the average base price per transaction is p, that would be different, but I think it's per item.Wait, the problem says: "the average base price of items purchased in these transactions is p." So, that could be interpreted as the average price per item is p. So, each item is p on average.Therefore, the calculations are correct.So, the total loss is 1125p.But let me express that as a fraction. 11.25 is 45/4, so 100*(45/4)p = (4500/4)p = 1125p. So, yes, that's consistent.Alternatively, 11.25p per transaction times 100 transactions is 1125p.Therefore, L = 1125p.But wait, let me make sure I didn't confuse the number of coupons or something. Each transaction has 15 items, each with 3 coupons, each coupon giving 25% off the base price. So, per item, the discount is 3*0.25p = 0.75p, so the loss per item is 0.75p, and per transaction, it's 15*0.75p = 11.25p.Yes, that's correct.So, summarizing:1. For the first part, each item is discounted by 75%, so the revenue per item is 0.25p, and for 10 items, it's 2.5p, which is (5/2)p.2. For the second part, each extreme couponer transaction causes a loss of 11.25p, and with 100 such transactions, the total loss is 1125p.Therefore, the answers are:1. R = (5/2)p2. L = 1125pBut let me write them in the required format.Final Answer1. The total revenue is boxed{dfrac{5}{2}p}.2. The total loss in revenue is boxed{1125p}.</think>

🧠:<think>Okay, so I have this problem about optimizing my music collection using matrices and vectors. Let me try to understand what it's asking for.First, there are 5 genres: Jazz, Classical, Rock, Pop, and Electronic. Each of these genres spans 4 different eras: 1950s, 1970s, 1990s, and 2010s. The influence of each genre from each era is given in a matrix A. The vector v represents the time I dedicate to each era proportionally. Part 1 asks me to calculate the weighted influence vector w by multiplying matrix A with vector v. So, I need to perform matrix multiplication here. Let me recall how matrix multiplication works. If A is a 5x4 matrix and v is a 4x1 vector, then the result w will be a 5x1 vector. Each element of w is the dot product of the corresponding row of A and the vector v.Let me write down the matrix A and vector v:Matrix A:3 2 5 14 3 6 25 4 7 36 5 8 47 6 9 5Vector v:0.20.30.40.1So, for each row in A, I need to multiply each element by the corresponding element in v and sum them up.Let me compute each component of w one by one.First component (Jazz):3*0.2 + 2*0.3 + 5*0.4 + 1*0.1= 0.6 + 0.6 + 2.0 + 0.1= 0.6 + 0.6 is 1.2, plus 2.0 is 3.2, plus 0.1 is 3.3Second component (Classical):4*0.2 + 3*0.3 + 6*0.4 + 2*0.1= 0.8 + 0.9 + 2.4 + 0.2= 0.8 + 0.9 is 1.7, plus 2.4 is 4.1, plus 0.2 is 4.3Third component (Rock):5*0.2 + 4*0.3 + 7*0.4 + 3*0.1= 1.0 + 1.2 + 2.8 + 0.3= 1.0 + 1.2 is 2.2, plus 2.8 is 5.0, plus 0.3 is 5.3Fourth component (Pop):6*0.2 + 5*0.3 + 8*0.4 + 4*0.1= 1.2 + 1.5 + 3.2 + 0.4= 1.2 + 1.5 is 2.7, plus 3.2 is 5.9, plus 0.4 is 6.3Fifth component (Electronic):7*0.2 + 6*0.3 + 9*0.4 + 5*0.1= 1.4 + 1.8 + 3.6 + 0.5= 1.4 + 1.8 is 3.2, plus 3.6 is 6.8, plus 0.5 is 7.3So, the weighted influence vector w is:w = (3.3, 4.3, 5.3, 6.3, 7.3)Let me double-check my calculations to make sure I didn't make any arithmetic errors.First component: 3*0.2=0.6, 2*0.3=0.6, 5*0.4=2.0, 1*0.1=0.1. Sum is 0.6+0.6=1.2, +2.0=3.2, +0.1=3.3. Correct.Second component: 4*0.2=0.8, 3*0.3=0.9, 6*0.4=2.4, 2*0.1=0.2. Sum is 0.8+0.9=1.7, +2.4=4.1, +0.2=4.3. Correct.Third component: 5*0.2=1.0, 4*0.3=1.2, 7*0.4=2.8, 3*0.1=0.3. Sum is 1.0+1.2=2.2, +2.8=5.0, +0.3=5.3. Correct.Fourth component: 6*0.2=1.2, 5*0.3=1.5, 8*0.4=3.2, 4*0.1=0.4. Sum is 1.2+1.5=2.7, +3.2=5.9, +0.4=6.3. Correct.Fifth component: 7*0.2=1.4, 6*0.3=1.8, 9*0.4=3.6, 5*0.1=0.5. Sum is 1.4+1.8=3.2, +3.6=6.8, +0.5=7.3. Correct.Alright, so part 1 seems done. Now, moving on to part 2.Part 2 says I want to balance the influences so that the total influence from each genre is evenly distributed. They define a balancing vector b such that when applied to w, the result is a vector where all elements are equal. Hmm, I need to figure out what operation is meant by "applied to." Since w is a vector, and b is a vector, maybe it's a multiplication? But w is 5x1 and b is...? If the result is a vector where all elements are equal, perhaps it's a scalar multiple.Wait, the wording is a bit unclear. It says, "when it is applied to the weighted influence vector w, the result is a vector where all elements are equal." So, maybe b is a vector such that w multiplied by b gives a vector with equal elements.But how? If w is 5x1 and b is 1x5, then their product would be a scalar. But the result is supposed to be a vector. Alternatively, if b is 5x5, then multiplying w by b would give a 5x1 vector. But the problem says "balancing vector b," which is likely a vector, not a matrix.Wait, perhaps it's component-wise multiplication? So, if we have w and b, both 5x1 vectors, then w .* b would be a 5x1 vector where each element is the product of the corresponding elements in w and b. Then, we want this resulting vector to have all elements equal.So, if I denote the resulting vector as c, then c = w .* b, and all elements of c are equal. Let me denote c as a vector where each element is k, some constant.Therefore, for each i, w_i * b_i = k.So, b_i = k / w_i.But since k is the same for all i, we can write b_i = k / w_i for each i. But we need to find b such that this holds. However, k is arbitrary, so we can set k to be 1, or any constant, but since b is a vector, perhaps we can set k such that the vector b is normalized in some way.Wait, but the problem doesn't specify any constraints on b, just that it's a vector that when applied to w, the result is a vector with all elements equal. So, perhaps b is a vector where each component is the reciprocal of w's components scaled by some constant.Let me think. If c is the resulting vector with all elements equal, say c = [k, k, k, k, k]^T, then we have:w .* b = cWhich implies:b = c ./ wBut since c is [k, k, k, k, k]^T, then:b_i = k / w_i for each i.But since k is arbitrary, we can set k to 1 for simplicity, so b_i = 1 / w_i.However, if we do that, then c would be [1,1,1,1,1]^T. But the problem says "the result is a vector where all elements are equal," so k can be any constant, not necessarily 1. So, perhaps we can set k to be the average of w, or something else.Wait, but if we set k to be the average of w, then c would be a vector where each element is the average of w. But in that case, b would be (average of w) divided by each w_i.Alternatively, maybe the problem wants the balancing vector b such that when you multiply w by b, you get a vector where all elements are equal to the same value, say k. So, b is a vector where each element is k divided by the corresponding element in w.But since k is arbitrary, perhaps we can set k to 1, which would make b = [1/w1, 1/w2, 1/w3, 1/w4, 1/w5]^T.But let me check the problem statement again: "the balancing vector b such that when it is applied to the weighted influence vector w, the result is a vector where all elements are equal."So, if we think of "applied to" as multiplication, it's ambiguous whether it's matrix multiplication or element-wise multiplication. Since w is a vector, and b is a vector, and the result is a vector, it's more likely element-wise multiplication.So, if we perform element-wise multiplication, then:w .* b = c, where c is a vector with all elements equal.Therefore, for each i, w_i * b_i = c_i, and all c_i are equal.Let me denote c_i = k for some constant k. Therefore, b_i = k / w_i.But since k is arbitrary, we can choose k such that b is a unit vector or something, but the problem doesn't specify. So, perhaps the simplest is to set k = 1, so b_i = 1 / w_i.But let me see if that makes sense. If I set b_i = 1 / w_i, then c_i = w_i * (1 / w_i) = 1 for all i. So, c would be [1,1,1,1,1]^T. That seems to satisfy the condition.But wait, is there another way? Maybe if we think of b as a diagonal matrix, but the problem says "balancing vector," which is a vector, not a matrix. So, probably element-wise multiplication.Alternatively, if we think of b as a row vector, and multiply w (a column vector) by b (a row vector), that would give a scalar, which is not a vector. So, that doesn't fit.Alternatively, if we think of b as a column vector, and perform matrix multiplication, but then w is 5x1 and b is 5x1, so their multiplication isn't defined unless we do w^T * b, which would be a scalar.Therefore, the only way to get a vector result is element-wise multiplication. So, I think that's the intended operation.Therefore, the balancing vector b is such that each component is the reciprocal of the corresponding component in w, scaled by a constant k. Since k can be any constant, we can set k=1 for simplicity, so b = [1/w1, 1/w2, 1/w3, 1/w4, 1/w5]^T.But let me check if this is the only solution. Suppose we have w .* b = c, where c is a vector of all k's. Then, b_i = k / w_i for each i. So, b is determined up to a scalar multiple k. So, the balancing vector is unique up to scaling.But the problem says "find the balancing vector b," so perhaps we can set k=1, making b = [1/w1, 1/w2, 1/w3, 1/w4, 1/w5]^T.Alternatively, if we want the resulting vector c to have a specific total, like the sum of c being equal to something, but the problem doesn't specify. It just says all elements are equal.So, I think the simplest solution is to set c as [1,1,1,1,1]^T, so b = [1/w1, 1/w2, 1/w3, 1/w4, 1/w5]^T.Given that, let's compute b.Given w = (3.3, 4.3, 5.3, 6.3, 7.3)So, b1 = 1 / 3.3 ≈ 0.3030b2 = 1 / 4.3 ≈ 0.2326b3 = 1 / 5.3 ≈ 0.1887b4 = 1 / 6.3 ≈ 0.1587b5 = 1 / 7.3 ≈ 0.1369So, b ≈ (0.3030, 0.2326, 0.1887, 0.1587, 0.1369)But let me write them as fractions to be precise.3.3 is 33/10, so 1/3.3 = 10/33 ≈ 0.30304.3 is 43/10, so 1/4.3 = 10/43 ≈ 0.23265.3 is 53/10, so 1/5.3 = 10/53 ≈ 0.18876.3 is 63/10, so 1/6.3 = 10/63 ≈ 0.15877.3 is 73/10, so 1/7.3 = 10/73 ≈ 0.1369So, the exact fractions are:b = (10/33, 10/43, 10/53, 10/63, 10/73)Alternatively, if we factor out 10, it's 10*(1/33, 1/43, 1/53, 1/63, 1/73). But since the problem doesn't specify any scaling, either form is acceptable.But let me check if this is the only way. Suppose instead of setting c to [1,1,1,1,1], we set c to some other constant vector, say [k,k,k,k,k]. Then, b would be (k/w1, k/w2, k/w3, k/w4, k/w5). So, b is a scalar multiple of (1/w1, 1/w2, 1/w3, 1/w4, 1/w5). Therefore, the balancing vector is unique up to scaling.But since the problem doesn't specify any particular scaling, we can choose k=1 for simplicity, making b = (1/w1, 1/w2, 1/w3, 1/w4, 1/w5).Alternatively, if we want the resulting vector c to have a specific total, like the sum of c being equal to the sum of w, but that's not specified.Wait, let me think again. The problem says "the result is a vector where all elements are equal." It doesn't specify the value, just that they are equal. So, the simplest solution is to set each element to 1, hence b = (1/w1, 1/w2, 1/w3, 1/w4, 1/w5).Therefore, the balancing vector b is:b = (10/33, 10/43, 10/53, 10/63, 10/73)Or approximately (0.3030, 0.2326, 0.1887, 0.1587, 0.1369)Let me verify this. If I multiply w by b element-wise:3.3 * 0.3030 ≈ 14.3 * 0.2326 ≈ 15.3 * 0.1887 ≈ 16.3 * 0.1587 ≈ 17.3 * 0.1369 ≈ 1Yes, each product is approximately 1, so c = [1,1,1,1,1]^T, which satisfies the condition.Alternatively, if I use the fractions:3.3 = 33/10, so 33/10 * 10/33 = 1Similarly for the others:43/10 * 10/43 = 153/10 * 10/53 = 163/10 * 10/63 = 173/10 * 10/73 = 1So, exact value is 1 for each component.Therefore, the balancing vector b is indeed (10/33, 10/43, 10/53, 10/63, 10/73).But let me see if there's another interpretation. Maybe the problem wants b such that when you multiply w by b (as in matrix multiplication, but that would require b to be a matrix). But since b is called a vector, it's more likely element-wise multiplication.Alternatively, if we think of b as a diagonal matrix, then w * b would be a diagonal matrix multiplied by w, but that would change the dimensions. Wait, no, if b is a diagonal matrix with elements b1 to b5 on the diagonal, then w * b would be a 5x5 matrix, which doesn't make sense because the result is supposed to be a vector.Alternatively, if b is a row vector, then w * b would be a scalar, which again doesn't fit.Therefore, the only feasible interpretation is element-wise multiplication, resulting in a vector where each element is equal.So, I think I've got it.Final Answer1. The weighted influence vector ( mathbf{w} ) is ( boxed{begin{pmatrix} 3.3 4.3 5.3 6.3 7.3 end{pmatrix}} ).2. The balancing vector ( mathbf{b} ) is ( boxed{begin{pmatrix} dfrac{10}{33} dfrac{10}{43} dfrac{10}{53} dfrac{10}{63} dfrac{10}{73} end{pmatrix}} ).</think>