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question:Andy is making fudge. First he needs to raise the temperature of the candy mixture from 60 degrees to 240 degrees. Then he needs to cool it down to x degrees. If the candy heats at 5 degrees/minute and cools at 7 degrees/minute, It will take 46 minutes for the candy to be done. What is the value of unknown variable x? What is the value of unknown variable x?
answer:To solve this problem, we need to determine the value of x, which represents the temperature to which the candy mixture needs to cool down. Let's break down the information given: Initial temperature: 60 degrees Final temperature before cooling: 240 degrees Cooling rate: 7 degrees/minute Total time for heating and cooling: 46 minutes First, let's calculate the time it takes for the candy mixture to heat up from 60 degrees to 240 degrees: Temperature increase: 240 - 60 = 180 degrees Heating rate: 5 degrees/minute Time for heating: 180 degrees / 5 degrees/minute = 36 minutes Now, let's calculate the remaining time for cooling: Total time: 46 minutes Time for heating: 36 minutes Remaining time for cooling: 46 minutes - 36 minutes = 10 minutes Since the cooling rate is 7 degrees/minute, we can calculate the temperature decrease during the remaining 10 minutes: Temperature decrease: 10 minutes * 7 degrees/minute = 70 degrees Therefore, the value of x, which represents the temperature to which the candy mixture needs to cool down, is 240 degrees - 70 degrees = 170 degrees. The value of x is 170. 170 The answer is: 170
question:The line passing through the points (-2,0) and (0,2) can be written in the form y = mx + b. What is the value of m + b?
answer:The slope of the line passing through the points (-2,0) and (0,2) can be found using the formula m = frac{y_2 - y_1}{x_2 - x_1}. Plugging in the coordinates, we get m = frac{2 - 0}{0 - (-2)} = frac{2}{2} = 1. To find the y-intercept, we can use the point-slope form of a line and plug in one of the points. Let's use (-2,0). The equation becomes 0 = 1(-2) + b, which simplifies to 0 = -2 + b. Solving for b, we get b = 2. Therefore, the equation of the line is y = x + 2. The value of m + b is 1 + 2 = boxed{3}.The answer is: 3
question:A circular spinner for a game has a radius of 5 cm. The probability of winning on one spin of this spinner is frac{2}{5}. What is the area, in sq cm, of the WIN sector? Express your answer in terms of pi. [asy]import graph; draw(Circle((0,0),25),black); draw((0,0)--(7,18),Arrow); draw((0,0)--(0,25)); draw((0,0)--(15,-20)); label("WIN",(10,10),S); label("LOSE",(-8,-8),N); dot((0,0)); [/asy]
answer:The probability of winning is equal to the ratio of the area of the WIN sector to the area of the entire circle. Since the radius of the spinner is 5 cm, the area of the entire circle is pi(5^2)=25pi sq cm. Let the area of the WIN sector be x sq cm. Then the probability of winning is frac{x}{25pi}=frac{2}{5}. Cross multiplying, we have 5x=50pi, so x=10pi. Therefore, the area of the WIN sector is boxed{10pi} sq cm. The answer is: 10pi
question:f the teacher brings in 14 mini-cupcakes and 12 donut holes for a class of 13 students, and each student receives an equal amount, how many desserts does each student receive?
answer:The teacher brings in a total of 14 mini-cupcakes + 12 donut holes = 26 desserts. There are 13 students in the class, so each student receives 26 desserts / 13 students = 2 desserts. Each student receives 2 desserts. 2 The answer is: 2