🧠:<think>Okay, so I have this problem about an auto enthusiast who has collected fuel efficiency data for ten cars. The fuel efficiencies are given as 15, 18, 12, 20, 16, 22, 14, 19, 21, and 17 km/L. There are two parts to this problem, and I need to solve both. Let me take them one by one.Problem 1: The first part says that the enthusiast wants to compare the fuel efficiency to an international standard given by the function ( f(x) = 0.5x^2 - 3x + 25 ), where ( x ) is the car's horsepower. He wants to find the necessary horsepower for each car to achieve exactly 20 km/L according to this function.Hmm, okay. So, for each car, we need to find the horsepower ( x ) such that ( f(x) = 20 ). That means we have to solve the equation ( 0.5x^2 - 3x + 25 = 20 ) for ( x ).Let me write that equation down:( 0.5x^2 - 3x + 25 = 20 )First, I should subtract 20 from both sides to set the equation to zero:( 0.5x^2 - 3x + 5 = 0 )Hmm, that's a quadratic equation. Quadratic equations are of the form ( ax^2 + bx + c = 0 ), so here, ( a = 0.5 ), ( b = -3 ), and ( c = 5 ).To solve this, I can use the quadratic formula:( x = frac{-b pm sqrt{b^2 - 4ac}}{2a} )Plugging in the values:( x = frac{-(-3) pm sqrt{(-3)^2 - 4 * 0.5 * 5}}{2 * 0.5} )Simplify step by step.First, compute the discriminant ( D = b^2 - 4ac ):( D = (-3)^2 - 4 * 0.5 * 5 = 9 - 10 = -1 )Wait, the discriminant is negative. That means there are no real solutions. Hmm, that can't be right because the problem is asking for necessary horsepower, implying there should be a real number solution.Let me double-check my calculations.Original equation: ( 0.5x^2 - 3x + 25 = 20 )Subtract 20: ( 0.5x^2 - 3x + 5 = 0 )Yes, that's correct.Quadratic formula: ( x = frac{-b pm sqrt{b^2 - 4ac}}{2a} )So, ( a = 0.5 ), ( b = -3 ), ( c = 5 )Discriminant: ( (-3)^2 - 4 * 0.5 * 5 = 9 - 10 = -1 )Hmm, negative discriminant. So, no real solutions. That suggests that according to this function, it's impossible for a car to have a fuel efficiency of exactly 20 km/L because the equation has no real roots.But wait, the function is given as ( f(x) = 0.5x^2 - 3x + 25 ). Let me see what the maximum fuel efficiency is. Since this is a quadratic function opening upwards (because the coefficient of ( x^2 ) is positive), it has a minimum point, not a maximum. So, the fuel efficiency can go to infinity as horsepower increases, but it has a minimum value.Wait, so the minimum fuel efficiency is at the vertex of the parabola. Let me compute the vertex.The vertex occurs at ( x = -b/(2a) ). Here, ( a = 0.5 ), ( b = -3 ).So, ( x = -(-3)/(2 * 0.5) = 3 / 1 = 3 ).So, at ( x = 3 ), the fuel efficiency is ( f(3) = 0.5*(9) - 3*(3) + 25 = 4.5 - 9 + 25 = 20.5 ) km/L.So, the minimum fuel efficiency is 20.5 km/L. Therefore, the function never goes below 20.5 km/L. So, 20 km/L is below the minimum, which is why the equation ( f(x) = 20 ) has no real solutions.Therefore, according to this function, it's impossible for a car to achieve exactly 20 km/L. So, the necessary horsepower doesn't exist for this fuel efficiency.But wait, the problem says "determine the necessary horsepower for each car to achieve a fuel efficiency of exactly 20 km/L according to the given function."Hmm, but if it's impossible, does that mean the answer is that there is no solution? Or maybe I made a mistake in interpreting the function.Wait, let me check the function again: ( f(x) = 0.5x^2 - 3x + 25 ). So, it's a quadratic function. Since the coefficient of ( x^2 ) is positive, it opens upwards, meaning it has a minimum point. So, the minimum fuel efficiency is 20.5 km/L at x=3, as calculated.Therefore, 20 km/L is less than the minimum, so it's impossible. So, there is no real horsepower that can achieve exactly 20 km/L.But the problem is asking for the necessary horsepower for each car. Wait, each car? So, does that mean for each of the ten cars, we need to find the horsepower x such that f(x) equals their respective fuel efficiencies?Wait, hold on. Maybe I misread the problem.Wait, the first part says: "determine the necessary horsepower for each car to achieve a fuel efficiency of exactly 20 km/L according to the given function."Wait, so it's not that each car has a different fuel efficiency, but all cars need to achieve 20 km/L. So, for each car, we need to find x such that f(x) = 20. But as we saw, this equation has no real solution, so for each car, it's impossible.Alternatively, maybe the problem is that each car has a different fuel efficiency, and we need to find the horsepower for each of their respective fuel efficiencies.Wait, let me read the problem again."If the auto enthusiast wants to compare the fuel efficiency of these cars to an international standard, which is given by the function ( f(x) = 0.5x^2 - 3x + 25 ), where ( x ) is the car's horsepower, determine the necessary horsepower for each car to achieve a fuel efficiency of exactly 20 km/L according to the given function."So, it says "for each car" to achieve exactly 20 km/L. So, regardless of their actual fuel efficiency, he wants to find the horsepower needed to get 20 km/L. But since the function's minimum is 20.5, which is higher than 20, it's impossible.Wait, but maybe I misread the function. Let me check again.Function: ( f(x) = 0.5x^2 - 3x + 25 ). So, yeah, that's correct.Alternatively, maybe the function is supposed to model fuel efficiency as a function of horsepower, so higher horsepower leads to lower fuel efficiency? But in this case, the function is quadratic with a positive coefficient on ( x^2 ), so as x increases, f(x) tends to infinity. So, higher horsepower leads to higher fuel efficiency, which doesn't make sense in real life. Usually, higher horsepower would mean lower fuel efficiency.Wait, maybe the function is given incorrectly? Or perhaps I misread it.Wait, the function is given as ( f(x) = 0.5x^2 - 3x + 25 ). So, if x is horsepower, then higher x would lead to higher fuel efficiency, which is counterintuitive. Maybe the function is supposed to be negative? Or perhaps it's a different model.Alternatively, maybe the function is correct, and it's just a hypothetical model where higher horsepower leads to higher fuel efficiency, which is not typical but possible in some contexts.But regardless, mathematically, the function has a minimum at x=3, with f(3)=20.5. So, 20 km/L is below that, so no solution.Therefore, for each car, there is no real horsepower x that would result in f(x)=20. So, the answer is that it's impossible.But the problem is asking to determine the necessary horsepower for each car. So, perhaps I need to explain that it's impossible because the function's minimum fuel efficiency is 20.5 km/L, which is higher than 20.Alternatively, maybe I made a mistake in the calculation.Wait, let me recalculate the discriminant.Equation: 0.5x² - 3x + 5 = 0Discriminant D = b² - 4ac = (-3)² - 4*(0.5)*5 = 9 - 10 = -1Yes, that's correct. So, D is negative, so no real solutions.Therefore, the answer is that there is no real horsepower x that satisfies f(x)=20.So, for each car, it's impossible to achieve exactly 20 km/L according to this function.But the problem says "determine the necessary horsepower for each car", so maybe I'm supposed to write that it's impossible for all of them.Alternatively, perhaps the function is supposed to be f(x) = -0.5x² - 3x + 25, which would open downward, having a maximum. Let me check.If the function was f(x) = -0.5x² - 3x + 25, then it would open downward, and the maximum fuel efficiency would be at x = -b/(2a) = -(-3)/(2*(-0.5)) = 3 / (-1) = -3. But horsepower can't be negative, so that doesn't make sense either.Alternatively, maybe the function is f(x) = 0.5x² + 3x + 25, but that would have a minimum at x = -b/(2a) = -3/(1) = -3, which is also negative.Alternatively, maybe the function is f(x) = -0.5x² + 3x + 25. Let's see.Then, the vertex is at x = -b/(2a) = -3/(2*(-0.5)) = -3 / (-1) = 3.f(3) = -0.5*(9) + 3*(3) +25 = -4.5 +9 +25= 30 -4.5=25.5So, maximum fuel efficiency is 25.5 km/L at x=3. Then, as x increases beyond 3, fuel efficiency decreases.In that case, solving f(x)=20 would have real solutions.But the original function given is ( f(x) = 0.5x^2 - 3x + 25 ). So, unless there's a typo, I have to go with that.Therefore, the conclusion is that it's impossible for any car to achieve exactly 20 km/L according to this function because the minimum fuel efficiency is 20.5 km/L.So, for each car, there is no solution.But the problem says "for each car", so maybe I need to write that for each car, it's impossible.Alternatively, maybe I misinterpreted the question.Wait, maybe the function is given as fuel efficiency as a function of horsepower, but the enthusiast wants to find the horsepower such that the fuel efficiency is 20 km/L. But since the function's minimum is 20.5, which is higher than 20, it's impossible.Therefore, the answer is that there is no real horsepower x that satisfies f(x)=20.So, for each car, it's impossible.But the problem is asking to determine the necessary horsepower for each car, so maybe I need to write that for each car, it's impossible.Alternatively, maybe the function is supposed to be f(x) = 0.5x² - 3x + 25, and the enthusiast wants to find the horsepower x such that f(x)=20, but since it's impossible, the answer is no solution.So, I think that's the case.Problem 2: The second part says that the fuel efficiencies correspond to cars with horsepowers modeled by the polynomial ( P(x) = 2x^3 - 5x^2 + 6x + 1 ), where ( x ) is the car index (starting from 1 for the first car, 2 for the second, etc.). We need to find the average horsepower of all ten cars.Okay, so for each car, we have an index x from 1 to 10, and the horsepower is given by P(x) = 2x³ -5x² +6x +1.So, to find the average horsepower, I need to compute P(1), P(2), ..., P(10), sum them all up, and divide by 10.That sounds straightforward, but computing P(x) for x from 1 to 10 might be time-consuming, but manageable.Alternatively, maybe there's a smarter way to compute the sum without calculating each term individually, but given that it's a cubic polynomial, summing it from x=1 to x=10 might require some formulas.Let me recall that the sum of x from 1 to n is n(n+1)/2.The sum of x² from 1 to n is n(n+1)(2n+1)/6.The sum of x³ from 1 to n is [n(n+1)/2]^2.Given that, since P(x) is 2x³ -5x² +6x +1, the sum of P(x) from x=1 to x=10 is 2*sum(x³) -5*sum(x²) +6*sum(x) + sum(1).So, let's compute each part.First, n=10.Compute sum(x) from 1 to 10: S1 = 10*11/2 = 55.Compute sum(x²) from 1 to 10: S2 = 10*11*21/6. Let's compute that.10*11=110, 110*21=2310, 2310/6=385.Compute sum(x³) from 1 to 10: S3 = [10*11/2]^2 = (55)^2 = 3025.Compute sum(1) from x=1 to 10: that's just 10.So, now, sum(P(x)) = 2*S3 -5*S2 +6*S1 +10.Plugging in the numbers:2*3025 = 6050-5*385 = -19256*55 = 330+10So, total sum = 6050 -1925 +330 +10.Compute step by step:6050 -1925 = 41254125 +330 = 44554455 +10 = 4465Therefore, the total horsepower is 4465.Average horsepower is total divided by 10: 4465 /10 = 446.5.So, the average horsepower is 446.5.But let me verify the calculations step by step to make sure.First, sum(x) = 55, correct.sum(x²) = 385, correct.sum(x³) = 3025, correct.sum(1) =10, correct.Then, 2*sum(x³) = 2*3025=6050.-5*sum(x²)= -5*385= -1925.6*sum(x)=6*55=330.sum(1)=10.Adding them up:6050 -1925 = 4125.4125 +330=4455.4455 +10=4465.Yes, that's correct.So, average is 4465 /10=446.5.Therefore, the average horsepower is 446.5.But let me think again. Is this correct?Alternatively, maybe I made a mistake in the polynomial.Wait, P(x)=2x³ -5x² +6x +1.So, when we sum P(x) from x=1 to 10, it's 2*sum(x³) -5*sum(x²) +6*sum(x) + sum(1). Yes, that's correct.So, the calculations seem correct.Therefore, the average horsepower is 446.5.But since horsepower is usually a whole number, maybe we need to round it? The problem doesn't specify, so 446.5 is fine.Alternatively, if we need to present it as a fraction, 446.5 is equal to 893/2.But probably, 446.5 is acceptable.So, summarizing:Problem 1: It's impossible for any car to achieve exactly 20 km/L according to the given function because the minimum fuel efficiency is 20.5 km/L.Problem 2: The average horsepower is 446.5.But wait, let me double-check the sum calculations.sum(x³) from 1 to 10 is 3025, correct.2*3025=6050.sum(x²)=385, 5*385=1925, so -5*385=-1925.sum(x)=55, 6*55=330.sum(1)=10.So, 6050 -1925=4125.4125 +330=4455.4455 +10=4465.Yes, that's correct.So, average is 4465/10=446.5.Yes, that seems correct.Therefore, the answers are:1. No real solution; it's impossible.2. Average horsepower is 446.5.But let me write them in the required format.For problem 1, since it's impossible, we can state that there is no real horsepower x that satisfies f(x)=20.For problem 2, the average is 446.5.So, I think that's it.Final Answer1. boxed{text{No solution}}2. boxed{446.5}</think>

🧠:<think>Okay, so I have two problems here about Process A and Process B in the textile industry. Let me try to figure them out step by step.Starting with Problem 1 about Process A. It involves sourcing raw materials from three countries: X, Y, and Z. Each country has different carbon emissions per metric ton of material. The emissions are 2, 3, and 1.5 metric tons from X, Y, and Z respectively. The fractions of raw material sourced from these countries are 0.4, 0.3, and 0.3. The total carbon emissions from shipping must not exceed 500 metric tons. I need to find the maximum amount of raw material that can be sourced.Hmm, okay. So, the total carbon emissions are a weighted average of the emissions from each country, based on the fraction sourced from each. So, if I let the total raw material be M metric tons, then the amount from X is 0.4M, from Y is 0.3M, and from Z is 0.3M.The carbon emissions from each country would then be:- From X: 2 * 0.4M- From Y: 3 * 0.3M- From Z: 1.5 * 0.3MAdding these up should give the total carbon emissions, which must be ≤ 500 metric tons.So, let me write that as an equation:Total emissions = (2 * 0.4M) + (3 * 0.3M) + (1.5 * 0.3M) ≤ 500Let me compute each term:2 * 0.4 = 0.8, so 0.8M3 * 0.3 = 0.9, so 0.9M1.5 * 0.3 = 0.45, so 0.45MAdding them together: 0.8M + 0.9M + 0.45M = (0.8 + 0.9 + 0.45)M = 2.15MSo, 2.15M ≤ 500To find M, divide both sides by 2.15:M ≤ 500 / 2.15Let me compute that. 500 divided by 2.15.First, 2.15 * 200 = 430Subtract 430 from 500: 70Now, 2.15 * 32 = 68.8So, 200 + 32 = 232, and 70 - 68.8 = 1.2So, 1.2 / 2.15 ≈ 0.558So, total M ≈ 232.558 metric tons.Since we can't have a fraction of a metric ton in this context, we might round down to 232 metric tons. But let me check the exact division.500 / 2.15 = ?Let me compute 500 / 2.15:2.15 goes into 500 how many times?2.15 * 200 = 430500 - 430 = 702.15 * 32 = 68.870 - 68.8 = 1.2So, 2.15 goes into 1.2 approximately 0.558 times.So, total is 200 + 32 + 0.558 ≈ 232.558So, approximately 232.56 metric tons.Since the problem doesn't specify rounding, maybe we can leave it as a decimal. So, M = 500 / 2.15 ≈ 232.558 metric tons.So, the maximum amount is approximately 232.56 metric tons.Wait, let me make sure I didn't make a calculation error.Alternatively, 2.15 * 232 = ?2 * 232 = 4640.15 * 232 = 34.8So, 464 + 34.8 = 498.8Which is just under 500. Then, 2.15 * 233 = 498.8 + 2.15 = 500.95, which is over 500.So, 232 metric tons would result in 498.8 metric tons of emissions, which is under 500. So, 232 is safe, but 233 would exceed.But the question is asking for the maximum amount, so if we can have a fraction, it's 232.56, but if we have to have a whole number, it's 232.But since it's metric tons, maybe fractions are acceptable. So, 232.56 is okay.So, I think the answer is approximately 232.56 metric tons.Moving on to Problem 2 about Process B. It focuses on minimizing overall costs, which include both shipping and manufacturing. The shipping costs are 50, 60, and 45 per metric ton from X, Y, and Z respectively. The manufacturing cost is 200 per metric ton of final product. The process requires 1.2 metric tons of raw material to produce 1 metric ton of final product. The total budget is 120,000. I need to find how many metric tons of the final product can be produced, assuming the same sourcing fractions as Process A.So, same sourcing fractions: 0.4 from X, 0.3 from Y, 0.3 from Z.First, let's figure out the cost per metric ton of raw material.The shipping cost per metric ton is a weighted average based on the fractions.So, shipping cost per metric ton:= (0.4 * 50) + (0.3 * 60) + (0.3 * 45)Let me compute each term:0.4 * 50 = 200.3 * 60 = 180.3 * 45 = 13.5Adding them up: 20 + 18 + 13.5 = 51.5So, shipping cost per metric ton of raw material is 51.50.Then, the manufacturing cost is 200 per metric ton of final product.But, it requires 1.2 metric tons of raw material to produce 1 metric ton of final product.So, for each metric ton of final product, the total cost is:Shipping cost for 1.2 metric tons + manufacturing cost.So, shipping cost for 1.2 metric tons: 1.2 * 51.50Let me compute that: 1.2 * 51.5051.50 * 1 = 51.5051.50 * 0.2 = 10.30Total: 51.50 + 10.30 = 61.80So, shipping cost per metric ton of final product is 61.80Then, manufacturing cost is 200 per metric ton.So, total cost per metric ton of final product is 61.80 + 200 = 261.80Given the total budget is 120,000, the number of metric tons produced is total budget divided by cost per metric ton.So, number of metric tons = 120,000 / 261.80Let me compute that.First, approximate 261.80 into 260 for easier calculation.120,000 / 260 ≈ 461.54But let's compute more accurately.261.80 * 458 = ?Wait, maybe do it step by step.Compute 120,000 / 261.80Let me write it as 120000 ÷ 261.80Let me compute 261.80 * 458:261.80 * 400 = 104,720261.80 * 50 = 13,090261.80 * 8 = 2,094.40Adding them up: 104,720 + 13,090 = 117,810; 117,810 + 2,094.40 = 119,904.40So, 261.80 * 458 = 119,904.40Subtract from 120,000: 120,000 - 119,904.40 = 95.60So, 95.60 / 261.80 ≈ 0.365So, total is approximately 458 + 0.365 ≈ 458.365So, approximately 458.37 metric tons.But let me check with another method.Alternatively, 261.80 * 458.365 ≈ 261.80 * 458 + 261.80 * 0.365Which we already saw is approximately 119,904.40 + 95.60 = 120,000.So, yes, 458.365 metric tons.But since we can't produce a fraction of a metric ton, we might have to round down to 458 metric tons.But let me confirm:Compute 458 * 261.80458 * 200 = 91,600458 * 60 = 27,480458 * 1.80 = 824.40Adding them up: 91,600 + 27,480 = 119,080; 119,080 + 824.40 = 119,904.40Then, 459 * 261.80 = 119,904.40 + 261.80 = 120,166.20, which is over the budget.So, 458 metric tons would cost 119,904.40, which is under the budget, and 459 would exceed.But the total budget is 120,000, so maybe we can produce 458 metric tons and have some money left.But the question is asking how many metric tons can be produced, so 458 is the maximum whole number.Alternatively, if partial metric tons are allowed, it's approximately 458.37.But in the context of production, it's likely they want a whole number, so 458 metric tons.Wait, but let me think again.The cost per metric ton is 261.80, so the number of metric tons is 120,000 / 261.80 ≈ 458.37.So, if they can produce 458.37 metric tons, but since you can't produce a fraction, it's 458.But sometimes, in budget terms, you might have some leftover money, but the question is about how many metric tons can be produced, so 458 is the answer.Wait, but let me check the exact calculation:120,000 / 261.80Let me compute this division precisely.261.80 goes into 120,000 how many times?First, 261.80 * 400 = 104,720Subtract from 120,000: 120,000 - 104,720 = 15,280Now, 261.80 * 50 = 13,090Subtract from 15,280: 15,280 - 13,090 = 2,190Now, 261.80 * 8 = 2,094.40Subtract from 2,190: 2,190 - 2,094.40 = 95.60So, total is 400 + 50 + 8 = 458, with a remainder of 95.60.So, 95.60 / 261.80 ≈ 0.365So, total is 458.365, which is approximately 458.37.So, if we can produce 458.37 metric tons, but since it's about production, it's probably 458 metric tons.But maybe the question expects the exact decimal, so 458.37.But let me check the question again: "how many metric tons of the final product can be produced, assuming the same sourcing fractions as Process A?"It doesn't specify rounding, so I think it's okay to present it as approximately 458.37 metric tons.But to be precise, since 458.37 is approximately 458.37, but in terms of exactness, it's 120,000 / 261.80.Alternatively, maybe we can represent it as a fraction.But 120,000 / 261.80 is equal to 120,000 / (26180/100) = 120,000 * 100 / 26180 = 12,000,000 / 26180.Simplify numerator and denominator by dividing numerator and denominator by 20: 600,000 / 1309.So, 600,000 ÷ 1309 ≈ 458.37.So, yeah, 458.37 metric tons.But since the question is about production, which is typically in whole numbers, maybe 458 metric tons.But perhaps the answer expects the exact decimal, so 458.37.I think both are acceptable, but since it's a budget, maybe they allow partial production, so 458.37.But let me see if I did everything correctly.First, shipping cost per metric ton of raw material is 0.4*50 + 0.3*60 + 0.3*45 = 20 + 18 + 13.5 = 51.5.Then, per metric ton of final product, you need 1.2 metric tons of raw material, so shipping cost is 1.2*51.5 = 61.8.Manufacturing cost is 200 per metric ton of final product.Total cost per metric ton of final product: 61.8 + 200 = 261.8.Total budget: 120,000.Number of metric tons: 120,000 / 261.8 ≈ 458.37.Yes, that seems correct.So, summarizing:Problem 1: Maximum raw material is approximately 232.56 metric tons.Problem 2: Maximum final product is approximately 458.37 metric tons.But let me just double-check the first problem.Total carbon emissions: 2.15M ≤ 500, so M ≤ 500 / 2.15 ≈ 232.558.Yes, that's correct.So, I think these are the answers.Final Answer1. The maximum amount of raw material that can be sourced is boxed{232.56} metric tons.2. The maximum amount of final product that can be produced is boxed{458.37} metric tons.</think>

🧠:<think>Okay, so Emily is planning this romantic trip through Europe, and she's got some math problems to solve. Let me try to figure them out step by step.First, problem 1: Emily is spending 10 days in France, visiting three landmarks—the Eiffel Tower, Palace of Versailles, and French Riviera. The probabilities of meeting her soulmate at each are 0.05, 0.02, and 0.03 respectively. We need to find the probability she meets her soulmate at least once during her visit to France. Hmm, okay.Since the events are independent, I think I can use the complement rule here. The probability of not meeting her soulmate at any of the landmarks is the product of the probabilities of not meeting at each one. So, for each landmark, the probability of not meeting is 1 minus the probability of meeting.So, for the Eiffel Tower, it's 1 - 0.05 = 0.95. For Versailles, 1 - 0.02 = 0.98. For the Riviera, 1 - 0.03 = 0.97. Then, multiplying these together gives the probability she doesn't meet her soulmate at any of the three. So, 0.95 * 0.98 * 0.97.Let me calculate that. 0.95 * 0.98 is... 0.95 * 0.98. Let's see, 0.95 * 1 is 0.95, so 0.95 * 0.98 is 0.95 - (0.95 * 0.02) = 0.95 - 0.019 = 0.931. Then, 0.931 * 0.97. Hmm, 0.931 * 1 is 0.931, so subtract 0.931 * 0.03. 0.931 * 0.03 is approximately 0.02793. So, 0.931 - 0.02793 = 0.90307.Therefore, the probability she doesn't meet her soulmate at any landmark is approximately 0.90307. So, the probability she meets at least once is 1 - 0.90307 = 0.09693, or about 9.693%.Wait, let me double-check that multiplication. Maybe I should calculate 0.95 * 0.98 first.0.95 * 0.98: 95 * 98. 95*100 is 9500, minus 95*2 is 190, so 9500 - 190 = 9310. So, 0.95 * 0.98 = 0.931. Then, 0.931 * 0.97.Let me compute 931 * 97. 931*100 is 93100, minus 931*3 is 2793. So, 93100 - 2793 = 90307. So, 0.931 * 0.97 = 0.90307. So, yeah, that's correct.So, 1 - 0.90307 is indeed 0.09693. So, approximately 9.69% chance she meets her soulmate at least once in France.Okay, moving on to problem 2: Emily is in Italy, inspired by the Fibonacci sequence. She visits three landmarks: Colosseum, Leaning Tower of Pisa, and Amalfi Coast. The number of days she spends at each follows the first three non-zero Fibonacci numbers, which are 1, 1, 2. So, she spends 1 day at Colosseum, 1 day at Pisa, and 2 days at Amalfi Coast.The cost per day at each landmark is given by the sequence a_n = 2^n, where n is the position in the sequence of her days spent. Wait, so does that mean for each landmark, the cost per day is 2^n, where n is the day number? Or is it the position in the sequence of days across all landmarks?Wait, the problem says, "the cost in euros of visiting each landmark per day is given by the sequence a_n = 2^n, where n is the position in the sequence of her days spent." Hmm. So, for each landmark, the cost per day is 2^n, where n is the day number within that landmark's visit.Wait, but she spends 1 day at Colosseum, 1 day at Pisa, and 2 days at Amalfi. So, for each landmark, the cost per day is 2^n, where n is the day within that landmark's visit.So, for Colosseum, she spends 1 day, so n=1, cost is 2^1 = 2 euros.For Pisa, also 1 day, n=1, cost is 2^1 = 2 euros.For Amalfi Coast, she spends 2 days. So, first day: n=1, cost=2^1=2 euros. Second day: n=2, cost=2^2=4 euros.Therefore, total cost for Amalfi Coast is 2 + 4 = 6 euros.So, total cost for Italy is Colosseum (2) + Pisa (2) + Amalfi (6) = 10 euros.Wait, that seems straightforward, but let me make sure I'm interpreting the problem correctly.The problem says: "the cost in euros of visiting each landmark per day is given by the sequence a_n = 2^n, where n is the position in the sequence of her days spent." So, for each landmark, the cost per day is 2^n, where n is the day within that landmark's visit.So, for each landmark, the first day is 2^1, second day 2^2, etc. So, for Colosseum, only 1 day: 2^1=2.Pisa: 1 day: 2^1=2.Amalfi: 2 days: 2^1 + 2^2 = 2 + 4 = 6.So, total cost: 2 + 2 + 6 = 10 euros.Alternatively, if the sequence was across all days, meaning the first day overall is 2^1, second day 2^2, etc., but that seems less likely because the problem says "per day is given by the sequence a_n = 2^n, where n is the position in the sequence of her days spent." So, it's per day per landmark.Wait, actually, the wording is a bit ambiguous. It says "the cost in euros of visiting each landmark per day is given by the sequence a_n = 2^n, where n is the position in the sequence of her days spent."So, perhaps for each landmark, the cost per day is 2^n, with n being the day number across all her visits. So, if she spends 1 day at Colosseum, 1 day at Pisa, and 2 days at Amalfi, then the days are ordered as Colosseum day 1, Pisa day 1, Amalfi day 1, Amalfi day 2.So, n=1: Colosseum day 1: 2^1=2.n=2: Pisa day 1: 2^2=4.n=3: Amalfi day 1: 2^3=8.n=4: Amalfi day 2: 2^4=16.So, total cost would be 2 + 4 + 8 + 16 = 30 euros.But that seems more expensive. Hmm.Wait, the problem says "the cost in euros of visiting each landmark per day is given by the sequence a_n = 2^n, where n is the position in the sequence of her days spent."So, it's per day, and n is the position in the sequence of her days spent. So, if she spends days in the order: Colosseum day 1, Pisa day 1, Amalfi day 1, Amalfi day 2, then n=1,2,3,4.So, the cost for each day is 2^1, 2^2, 2^3, 2^4.Therefore, total cost is 2 + 4 + 8 + 16 = 30 euros.But wait, the problem says "the number of days she plans to spend at each landmark follows the first three non-zero Fibonacci numbers (1, 1, 2)." So, the days are split as 1,1,2. So, the sequence of days is 1,1,2, but the total days are 4.But the cost per day is 2^n, where n is the position in the sequence of her days spent. So, if the days are ordered as 1 (Colosseum), 2 (Pisa), 3 (Amalfi), 4 (Amalfi), then n=1,2,3,4.So, each day's cost is 2^1, 2^2, 2^3, 2^4, which totals 2 + 4 + 8 + 16 = 30.Alternatively, if the days are considered per landmark, meaning for each landmark, the days are counted separately, then for Colosseum: 1 day, cost=2^1=2.Pisa: 1 day, cost=2^1=2.Amalfi: 2 days, cost=2^1 + 2^2=2 + 4=6.Total: 2 + 2 + 6=10.So, which interpretation is correct? The problem says "the cost in euros of visiting each landmark per day is given by the sequence a_n = 2^n, where n is the position in the sequence of her days spent."So, "per day" and "position in the sequence of her days spent." So, the sequence is across all days, not per landmark.Therefore, the first day she spends anywhere is n=1, second day n=2, etc.So, since she spends 1 day at Colosseum, 1 day at Pisa, and 2 days at Amalfi, the total days are 4.Therefore, the costs are 2^1, 2^2, 2^3, 2^4, summing to 30.But let me think again. If it's per landmark, then each landmark's days are separate. So, for each landmark, the first day is n=1, second day n=2, etc.So, for Colosseum: 1 day, cost=2^1=2.For Pisa: 1 day, cost=2^1=2.For Amalfi: 2 days, cost=2^1 + 2^2=2 + 4=6.Total: 2 + 2 + 6=10.But the problem says "the position in the sequence of her days spent." So, if she spends days in the order: Colosseum day 1, Pisa day 1, Amalfi day 1, Amalfi day 2, then the sequence is 1,2,3,4.So, n=1: 2^1=2.n=2: 2^2=4.n=3: 2^3=8.n=4: 2^4=16.Total: 2 + 4 + 8 + 16=30.Hmm, this is a bit ambiguous. The problem could be interpreted either way. But I think the key is that it's the "sequence of her days spent," meaning the overall sequence of days across all landmarks. So, the first day is day 1, second day day 2, etc., regardless of which landmark she's at.Therefore, the total cost would be 2 + 4 + 8 + 16=30 euros.But let me see if there's another way. Maybe for each landmark, the days are counted separately, so for each landmark, the first day is n=1, second day n=2, etc., but across all landmarks, the sequence is reset for each landmark.So, for Colosseum: 1 day, cost=2^1=2.For Pisa: 1 day, cost=2^1=2.For Amalfi: 2 days, cost=2^1 + 2^2=2 + 4=6.Total: 10.I think this is the correct interpretation because the problem says "the cost in euros of visiting each landmark per day is given by the sequence a_n = 2^n, where n is the position in the sequence of her days spent."So, for each landmark, the days are separate. So, for each landmark, the first day is n=1, second day n=2, etc. So, the sequence is per landmark, not globally.Therefore, the total cost is 2 + 2 + 6=10 euros.Wait, but the problem says "the position in the sequence of her days spent." So, if she spends days in the order: Colosseum day 1, Pisa day 1, Amalfi day 1, Amalfi day 2, then the position in the sequence is 1,2,3,4.So, the cost for each day is 2^1, 2^2, 2^3, 2^4.Therefore, total cost is 2 + 4 + 8 + 16=30.I think this is the correct interpretation because the sequence is of her days spent, meaning the overall sequence, not per landmark.So, the answer is 30 euros.Wait, but the problem says "visiting each landmark per day." So, maybe for each landmark, the cost per day is 2^n, where n is the day within that landmark's visit.So, for Colosseum: 1 day, n=1: 2^1=2.Pisa: 1 day, n=1: 2^1=2.Amalfi: 2 days, n=1 and n=2: 2 + 4=6.Total: 10.I think this is the correct interpretation because it says "visiting each landmark per day," meaning per day at that landmark. So, each landmark's days are separate.Therefore, the total cost is 10 euros.But I'm still a bit confused. Let me try to parse the sentence again: "the cost in euros of visiting each landmark per day is given by the sequence a_n = 2^n, where n is the position in the sequence of her days spent."So, "visiting each landmark per day" – so for each day she spends at a landmark, the cost is 2^n, where n is the position in the sequence of her days spent.So, if she spends days in the order: Colosseum day 1, Pisa day 1, Amalfi day 1, Amalfi day 2, then n=1,2,3,4.Therefore, the cost for each day is 2^1, 2^2, 2^3, 2^4.So, total cost is 2 + 4 + 8 + 16=30.Yes, I think that's the correct interpretation because the sequence is of her days spent, meaning the overall sequence, not per landmark.Therefore, the total cost is 30 euros.Wait, but the problem says "visiting each landmark per day." So, maybe it's per day per landmark, meaning for each day at a landmark, the cost is 2^n, where n is the day within that landmark's visit.So, for Colosseum: 1 day, n=1: 2.Pisa: 1 day, n=1: 2.Amalfi: 2 days, n=1 and n=2: 2 + 4=6.Total: 10.I think this is the correct interpretation because it's per day per landmark, so the sequence is per landmark.Therefore, the total cost is 10 euros.But I'm still not entirely sure. The problem is a bit ambiguous. However, given the wording "the position in the sequence of her days spent," it seems like it's the overall sequence, not per landmark.So, I think the total cost is 30 euros.Wait, but let me think about the Fibonacci part. The number of days she spends at each landmark follows the first three non-zero Fibonacci numbers: 1,1,2. So, the days are 1,1,2, totaling 4 days.Therefore, the sequence of her days spent is 4 days. So, n=1,2,3,4.Thus, the cost per day is 2^1, 2^2, 2^3, 2^4, which are 2,4,8,16.So, total cost is 2+4+8+16=30.Yes, that makes sense.Therefore, the total cost is 30 euros.So, to summarize:Problem 1: Probability Emily meets her soulmate at least once in France is approximately 9.69%.Problem 2: Total cost of her visits in Italy is 30 euros.Final Answer1. The probability Emily meets her soulmate at least once in France is boxed{0.09693}.2. The total cost of her visits in Italy is boxed{30} euros.</think>

🧠:<think>Okay, so I have this problem about a German basketball player who wants to model his jump shot with a quadratic function. There are two parts to the problem. Let me try to tackle them one by one.Starting with part 1: I need to find the initial vertical velocity ( v_0 ) required for the ball to reach a maximum height of 3.5 meters. The equation given is ( h(t) = -4.9t^2 + v_0 t + h_0 ). I know that ( h_0 ) is the initial height of the ball, which is the player's height plus 0.5 meters. The player is 210 cm tall, so that's 2.1 meters. Therefore, ( h_0 = 2.1 + 0.5 = 2.6 ) meters.Now, the maximum height of a parabola given by ( h(t) = at^2 + bt + c ) occurs at ( t = -frac{b}{2a} ). In this case, ( a = -4.9 ) and ( b = v_0 ). So, the time at which the maximum height occurs is ( t = -frac{v_0}{2(-4.9)} = frac{v_0}{9.8} ).To find the maximum height, I plug this time back into the height equation:( h_{max} = -4.9 left( frac{v_0}{9.8} right)^2 + v_0 left( frac{v_0}{9.8} right) + 2.6 )Simplifying this:First, square ( frac{v_0}{9.8} ):( left( frac{v_0}{9.8} right)^2 = frac{v_0^2}{96.04} )Multiply by -4.9:( -4.9 times frac{v_0^2}{96.04} = -frac{4.9}{96.04} v_0^2 )Simplify ( frac{4.9}{96.04} ). Let me calculate that:4.9 divided by 96.04. Hmm, 4.9 is approximately 0.05098 times 96.04. Wait, actually, 4.9 divided by 96.04 is approximately 0.05098. So, ( -0.05098 v_0^2 ).Then, the second term is ( v_0 times frac{v_0}{9.8} = frac{v_0^2}{9.8} approx 0.10204 v_0^2 ).Adding these together:( -0.05098 v_0^2 + 0.10204 v_0^2 = 0.05106 v_0^2 )So, the maximum height equation becomes:( h_{max} = 0.05106 v_0^2 + 2.6 )We are told that the maximum height needs to be 3.5 meters. So:( 3.5 = 0.05106 v_0^2 + 2.6 )Subtract 2.6 from both sides:( 3.5 - 2.6 = 0.05106 v_0^2 )( 0.9 = 0.05106 v_0^2 )Now, solve for ( v_0^2 ):( v_0^2 = frac{0.9}{0.05106} )Calculating that:0.9 divided by 0.05106. Let me see, 0.05106 times 17.6 is approximately 0.9 (since 0.05106*17=0.868, 0.05106*18=0.919). So, approximately 17.6.Therefore, ( v_0^2 approx 17.6 ), so ( v_0 approx sqrt{17.6} ).Calculating square root of 17.6: 4.2 squared is 17.64, so it's approximately 4.2 m/s.Wait, let me check my calculations again because I approximated some numbers.Starting from:( h_{max} = -4.9 left( frac{v_0}{9.8} right)^2 + v_0 left( frac{v_0}{9.8} right) + 2.6 )Let me compute this without approximating:First term: ( -4.9 times frac{v_0^2}{96.04} = -frac{4.9}{96.04} v_0^2 ). 4.9 divided by 96.04 is exactly 0.050980392.Second term: ( frac{v_0^2}{9.8} = 0.102040816 v_0^2 ).So, combining the two:( -0.050980392 v_0^2 + 0.102040816 v_0^2 = (0.102040816 - 0.050980392) v_0^2 = 0.051060424 v_0^2 ).So, ( h_{max} = 0.051060424 v_0^2 + 2.6 ).Set equal to 3.5:( 0.051060424 v_0^2 = 0.9 )Therefore, ( v_0^2 = 0.9 / 0.051060424 approx 17.62 ).So, ( v_0 = sqrt{17.62} approx 4.198 ) m/s.Rounding to two decimal places, that's approximately 4.20 m/s.So, the initial vertical velocity ( v_0 ) is about 4.20 m/s.Wait, let me double-check if I did everything correctly. The maximum height occurs at ( t = v_0 / 9.8 ). Plugging that into the height equation:( h(t) = -4.9 (v_0^2 / 96.04) + v_0 (v_0 / 9.8) + 2.6 ).Yes, that's correct. So, the calculation seems right.Moving on to part 2: The player wants to score a 3-point shot from 6.75 meters away. The horizontal distance is given by ( x(t) = u_0 t ), where ( u_0 ) is the initial horizontal velocity. The ball must reach the basket at a height of 3.05 meters when it's 6.75 meters away.So, we need to find ( u_0 ) such that when ( x(t) = 6.75 ), ( h(t) = 3.05 ).First, let's note that at the time when the ball reaches the basket, both the horizontal and vertical positions must satisfy their respective equations.From the horizontal equation: ( x(t) = u_0 t ). So, ( t = 6.75 / u_0 ).From the vertical equation: ( h(t) = -4.9 t^2 + v_0 t + h_0 = 3.05 ).We already found ( v_0 ) in part 1, which is approximately 4.20 m/s, and ( h_0 = 2.6 ) meters.So, substituting ( t = 6.75 / u_0 ) into the vertical equation:( 3.05 = -4.9 left( frac{6.75}{u_0} right)^2 + 4.20 left( frac{6.75}{u_0} right) + 2.6 )Let me write this equation out:( 3.05 = -4.9 left( frac{45.5625}{u_0^2} right) + 4.20 left( frac{6.75}{u_0} right) + 2.6 )Simplify each term:First term: ( -4.9 times 45.5625 / u_0^2 = -223.25625 / u_0^2 )Second term: ( 4.20 times 6.75 / u_0 = 28.35 / u_0 )Third term: 2.6So, putting it all together:( 3.05 = -223.25625 / u_0^2 + 28.35 / u_0 + 2.6 )Subtract 3.05 from both sides:( 0 = -223.25625 / u_0^2 + 28.35 / u_0 + 2.6 - 3.05 )Simplify the constants:2.6 - 3.05 = -0.45So, equation becomes:( 0 = -223.25625 / u_0^2 + 28.35 / u_0 - 0.45 )Multiply both sides by ( u_0^2 ) to eliminate denominators:( 0 = -223.25625 + 28.35 u_0 - 0.45 u_0^2 )Rearranging terms:( -0.45 u_0^2 + 28.35 u_0 - 223.25625 = 0 )Multiply both sides by -1 to make the quadratic coefficient positive:( 0.45 u_0^2 - 28.35 u_0 + 223.25625 = 0 )Now, this is a quadratic equation in terms of ( u_0 ):( 0.45 u_0^2 - 28.35 u_0 + 223.25625 = 0 )Let me write it as:( 0.45 u_0^2 - 28.35 u_0 + 223.25625 = 0 )To make it easier, let's multiply all terms by 1000 to eliminate decimals:( 450 u_0^2 - 28350 u_0 + 223256.25 = 0 )Hmm, that might not be necessary. Alternatively, let's use the quadratic formula.Quadratic formula: ( u_0 = frac{-b pm sqrt{b^2 - 4ac}}{2a} )Where ( a = 0.45 ), ( b = -28.35 ), ( c = 223.25625 ).Compute discriminant ( D = b^2 - 4ac ):( D = (-28.35)^2 - 4 * 0.45 * 223.25625 )Calculate each part:First, ( (-28.35)^2 = 803.7225 )Second, ( 4 * 0.45 = 1.8 ), then ( 1.8 * 223.25625 = 401.86125 )So, ( D = 803.7225 - 401.86125 = 401.86125 )So, discriminant is positive, which is good.Now, square root of D: ( sqrt{401.86125} approx 20.0465 )So, ( u_0 = frac{-(-28.35) pm 20.0465}{2 * 0.45} = frac{28.35 pm 20.0465}{0.9} )Compute both possibilities:First, with the plus sign:( u_0 = frac{28.35 + 20.0465}{0.9} = frac{48.3965}{0.9} approx 53.7739 ) m/sSecond, with the minus sign:( u_0 = frac{28.35 - 20.0465}{0.9} = frac{8.3035}{0.9} approx 9.2261 ) m/sNow, we need to consider which of these solutions makes sense. A horizontal velocity of 53.77 m/s seems extremely high for a basketball shot. Professional players can throw the ball at around 10-15 m/s, so 53 m/s is unrealistic. On the other hand, 9.23 m/s seems more plausible.Therefore, the required initial horizontal velocity ( u_0 ) is approximately 9.23 m/s.But let me verify this because sometimes when dealing with quadratics, both solutions might be mathematically correct but only one is physically meaningful.So, if ( u_0 = 9.23 ) m/s, then the time ( t = 6.75 / 9.23 approx 0.731 ) seconds.Let's plug this time into the vertical equation:( h(t) = -4.9 (0.731)^2 + 4.20 (0.731) + 2.6 )Calculate each term:First term: ( -4.9 * 0.534 approx -2.6166 )Second term: ( 4.20 * 0.731 approx 3.0702 )Third term: 2.6Adding them together: -2.6166 + 3.0702 + 2.6 ≈ 3.0536 meters, which is approximately 3.05 meters. That checks out.If we use the other solution, ( u_0 = 53.77 ) m/s, then ( t = 6.75 / 53.77 ≈ 0.1255 ) seconds.Plugging into vertical equation:( h(t) = -4.9 (0.1255)^2 + 4.20 (0.1255) + 2.6 )First term: ( -4.9 * 0.01575 ≈ -0.0772 )Second term: ( 4.20 * 0.1255 ≈ 0.5271 )Third term: 2.6Total: -0.0772 + 0.5271 + 2.6 ≈ 3.05 meters. So, mathematically, both solutions work.But as I thought earlier, 53.77 m/s is way too high for a basketball shot. Professional players can't throw the ball that fast. The maximum horizontal velocity for a basketball shot is typically around 10-15 m/s. So, 9.23 m/s is the realistic solution.Therefore, the required initial horizontal velocity ( u_0 ) is approximately 9.23 m/s.Wait, let me see if I can express this more accurately. The quadratic solution gave me approximately 9.2261 m/s, which is about 9.23 m/s when rounded to two decimal places.So, summarizing:1. The initial vertical velocity ( v_0 ) is approximately 4.20 m/s.2. The initial horizontal velocity ( u_0 ) is approximately 9.23 m/s.I think that's it. Let me just recap to make sure I didn't miss anything.For part 1, I correctly identified the maximum height formula, plugged in the known values, solved for ( v_0 ), and got a reasonable value.For part 2, I set up the equations for both horizontal and vertical motion, substituted the time from the horizontal equation into the vertical equation, solved the resulting quadratic, and selected the physically meaningful solution.Everything seems to check out. I don't see any calculation errors in my steps, and the results make sense in the context of basketball physics.Final Answer1. The required initial vertical velocity is boxed{4.20} m/s.2. The required initial horizontal velocity is boxed{9.23} m/s.</think>