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question:Every year 20 ducks from a flock are killed but another 30 are born. The original flock size is 100 ducks. After 5 years they join with another flock of 150 ducks. How many ducks are in the combined flock?
answer:Every year, the flock loses 20 ducks and gains 30 ducks, so the net increase each year is 30 - 20 = 10 ducks. After 5 years, the flock will have increased by 10 ducks/year x 5 years = 50 ducks. The original flock size is 100 ducks, so after 5 years, there will be 100 + 50 = 150 ducks in the original flock. When they join with another flock of 150 ducks, the combined total is 150 + 150 = 300 ducks. 300 The answer is: 300
question:Let S be the set of all real values of x with X < x < frac{pi}{2} such that sin x, cos x, and tan x form the side lengths (in some order) of a right triangle. Compute the sum of tan^2 x over all x in S. The answer is 2. What is the value of unknown variable X?
answer:es that the sum of tan^2 x over all x in S is 2. Therefore, we can set up the equation as follows: tan^2 x + tan^2 (frac{pi}{2} - x) + tan^2 (frac{pi}{2} + x) = 2 Since tan (frac{pi}{2} - x) = frac{1}{tan x} and tan (frac{pi}{2} + x) = -cot x, we can rewrite the equation as: tan^2 x + frac{1}{tan^2 x} + (-cot^2 x) = 2 To simplify the equation, we can use the identity cot^2 x = frac{1}{tan^2 x}: tan^2 x + frac{1}{tan^2 x} + frac{1}{tan^2 x} = 2 Combining like terms, we get: tan^2 x + frac{2}{tan^2 x} = 2 Multiplying both sides of the equation by tan^2 x, we have: (tan^2 x)^2 + 2 = 2(tan^2 x) Let y = tan^2 x. The equation becomes: y^2 + 2 = 2y Rearranging the equation, we get: y^2 - 2y + 2 = 0 This quadratic equation does not have real solutions. Therefore, there is no value of the unknown variable x that satisfies the given equation. The answer is: 0
question:Nancy can hula hoop for 10 minutes. Casey can hula hoop 3 minutes less than Nancy. Morgan can hula hoop three times as long as Casey. How long can Morgan hula hoop?
answer:Nancy can hula hoop for 10 minutes. Casey can hula hoop 3 minutes less than Nancy, so Casey can hula hoop for 10 - 3 = 7 minutes. Morgan can hula hoop three times as long as Casey, so Morgan can hula hoop for 7 * 3 = 21 minutes. Therefore, Morgan can hula hoop for 21 minutes. 21 The answer is: 21
question:The price of electricity went up by 25%. John's old computer used 800 watts and his new computer uses 50% more. If the old price of electricity was 12 cents per kilowatt-hour, how much does his computer cost, in dollars, to run for 50 hours?
answer:The new computer uses 800 watts * 1.5 = 1200 watts. To convert watts to kilowatts, we divide by 1000, so the new computer uses 1200 / 1000 = 1.2 kilowatts. The price of electricity went up by 25%, so the new price is 12 cents * (1 + 25%) = 12 cents * 1.25 = 15 cents per kilowatt-hour. To find the cost to run the new computer for 50 hours, we multiply the power usage in kilowatts by the price per kilowatt-hour and by the number of hours: 1.2 kilowatts * 15 cents/kilowatt-hour * 50 hours = 900 cents. To convert cents to dollars, we divide by 100, so the cost is 900 cents / 100 = 9. 9 The answer is: 9