MIT

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question:Lilith originally had five dozen water bottles that she needed to sell at 2 each to get exactly enough money to buy her friend a birthday gift. However, at the store, Lilith realized she could not sell at 2 because the regular price was x per water bottle in her town, and she had to reduce her price to 1.85 as well to sell her water bottles. Calculate the total amount of money Lilith will have to find to buy her friend the birthday gift after selling her water bottles at the reduced cost. If we know the answer to the above question is 9, what is the value of unknown variable x?

answer:Lilith originally had 5 dozen water bottles, which is equal to 5 * 12 = 60 water bottles. She planned to sell each water bottle for 2, but she had to reduce the price to 1.85. The total amount of money she would have gotten if she sold all the water bottles at 2 each is 60 * 2 = 120. However, she sold them at 1.85 each, so the total amount of money she actually got is 60 * 1.85 = 111. We are given that Lilith will have to find a total of 9 to buy her friend the birthday gift, so we can write: 111 - 9 = 102. We are also given that the regular price of a water bottle is x, so we can write: 1.85 = x. Solving for x, we get: x = 1.85. The value of x is 1.85. 1.85 The answer is: 1.85

question:What is the largest possible value of d in the 6-digit multiples of 22 in the form 5d5,!22e, where d and e are digits?

answer:We know that a number is divisible by 22 if and only if it is divisible by 2 and 11. Since the number is even, the units digit e must be even. Since the sum of the digits must be divisible by 11, we know that d + 5 + 5 + 2 + 2 + e is divisible by 11. Simplifying, we have d + e + 14 is divisible by 11. Since d and e are digits, d + e + 14 must be less than or equal to 9 + 9 + 14 = 32. We want to maximize d, so we want to maximize d + e + 14. The largest multiple of 11 less than or equal to 32 is 22, so d + e + 14 = 22. Solving for d, we have d = 8 - e. Since d is a digit, the largest possible value for d is boxed{8}. The answer is: 8

question:At what value of y will there be a horizontal asymptote for the equation y = frac{12x^4+4x^3+9x^2+5x+3}{3x^4+2x^3+8x^2+3x+1}?

answer:For there to be a horizontal asymptote, the degree of the numerator must be equal to or less than the degree of the denominator. The degree of the numerator is 4 and the degree of the denominator is also 4, so there is a possibility of a horizontal asymptote. To find the value of y at the horizontal asymptote, we divide the leading terms of the numerator and denominator. The leading term of the numerator is 12x^4 and the leading term of the denominator is 3x^4. When we divide these leading terms, we get frac{12x^4}{3x^4} = 4. Therefore, the horizontal asymptote occurs at y = boxed{4}. The answer is: 4

question:Let [f(x) = begin{cases} 2x + 9 &text{if }x<-2, 5-2x&text{if }xge -2. end{cases} ]Find f(3).

answer:Since 3 ge -2, we use the second case to find that f(3) = 5 - 2(3) = 5 - 6 = boxed{-1}. The answer is: -1