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question:During the car ride home, Michael looks back at his recent math exams. A problem on Michael's calculus mid-term gets him starting thinking about a particular quadratic,[x^2-sx+p,]with roots r_1 and r_2. He notices that[r_1+r_2=r_1^2+r_2^2=r_1^3+r_2^3=cdots=r_1^{2007}+r_2^{2007}.]He wonders how often this is the case, and begins exploring other quantities associated with the roots of such a quadratic. He sets out to compute the greatest possible value of[dfrac1{r_1^{2008}}+dfrac1{r_2^{2008}}.]Help Michael by computing this maximum.
answer:et P = r_1 r_2. We are given that S = S^2 = S^3 = dotsb = S^{2007}. We are asked to find the maximum value of frac{1}{r_1^{2008}} + frac{1}{r_2^{2008}} = frac{r_1^{2008} + r_2^{2008}}{(r_1 r_2)^{2008}} = frac{(r_1^{2008} + r_2^{2008})(r_1^{2008} + r_2^{2008})}{(r_1^{2008} + r_2^{2008})(r_1^{2008} + r_2^{2008}) (r_1^{2008} + r_2^{2008}) dotsm (r_1^{2008} + r_2^{2008})} = frac{(S^{2008} - 2P^{2008})}{(S^{2008})(S^{2008} - P^{2008})}. Since S = S^2, it follows that S = 0 or S = 1. If S = 0, then we have frac{0^{2008} - 2P^{2008}}{0^{2008}(0^{2008} - P^{2008})} = frac{2P^{2008}}{(P^{2008})} = 2. If S = 1, then we have frac{1^{2008} - 2P^{2008}}{1^{2008}(1^{2008} - P^{2008})} = frac{1 - 2P^{2008}}{1 - P^{2008}}. We want to maximize this fraction, so we want to minimize the denominator. Since P = r_1 r_2, it follows that P ge 0. Hence, P^{2008} ge 0, so we have frac{1 - 2P^{2008}}{1 - P^{2008}} le frac{1}{1} = 1. Therefore, the maximum value of frac{1}{r_1^{2008}} + frac{1}{r_2^{2008}} is boxed{2}.The answer is: 2
question:Lori owns a carsharing company and has three red cars and two white cars available for rent. The rental rate for the white car is 2 per minute, and for the red car, it is 3 per minute. All cars were rented for a duration of 3 hours. How much money did Lori earn in total?
answer:Lori has 3 red cars and 2 white cars, so she had a total of 3 + 2 = 5 cars available for rent. The duration of the rental was 3 hours, which is equal to 3 * 60 = 180 minutes. For the white cars, the rental rate is 2 per minute, so for each white car, Lori earned 2 * 180 = 360. For the red cars, the rental rate is 3 per minute, so for each red car, Lori earned 3 * 180 = 540. Since Lori had 3 red cars, she earned a total of 3 * 540 = 1620 from the red cars. And since Lori had 2 white cars, she earned a total of 2 * 360 = 720 from the white cars. Therefore, Lori earned a total of 1620 + 720 = 2340 in total. 2340 The answer is: 2340
question:Consecutive powers of X are added to form this sequence: 3^0,3^0+ 3^1, 3^0+ 3^1+ 3^2, and so on. The simplified value of the fourth term of the sequence is 40. What is the value of unknown variable X?
answer:To solve this problem, we need to determine the value of x, which represents the base of the powers in the sequence. Let's break down the information given: Simplified value of the fourth term of the sequence: 40 We can set up the equation as follows: 3^0 + 3^1 + 3^2 + x^3 = 40 Let's simplify and solve for x: 1 + 3 + 9 + x^3 = 40 13 + x^3 = 40 To isolate x^3, we subtract 13 from both sides of the equation: 13 + x^3 - 13 = 40 - 13 x^3 = 27 To solve for x, we take the cube root of both sides of the equation: ∛(x^3) = ∛27 x = 3 The value of x is 3. The answer is: 3
question:What is the sum of all the prime numbers between 1 and 10?
answer:The prime numbers between 1 and 10 are 2, 3, 5, and 7. The sum of these prime numbers is 2 + 3 + 5 + 7 = boxed{17}. The answer is: 17