Question 1061758: Over a specified distance, rate varies inversely with time. If a car on a test track goes a certain distance in one dash one-half minute at 150 mph, what rate is needed to go the same distance in two dash two-thirds minute?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! formula for inverse variation is:
y = k / x
k is the constant of variation.
in this problem k will represent the distance, which will be constant.
in general, you should convert everything to the same units of measure so as to be consistent.
in this problem, it didn't matter, but i will do the conversion anyway just to be consistent with the general rule, and to find the actual distance accurately.
first we'll do the conversion.
1 and 1/2 minutes is equal to 3/2 minutes divided by 60 which is equal to 3/120 hours which is equal to 1/40 hours.
2 and 2/3 minutes is equal to 8/3 minutes divided by 60 which is equal to 8 / 180 hours which is equal to 2/45 hours.
if we let k equal the distance and y equal the miles per hour and x equal the time, then the formula of y = k/x becomes rate = distance / time.
when rate = 150 mph and time = 1/40 hours, we get 150 = distance / (1/40).
we solve for distance to get distance = 150 / 40 = 3.75 miles.
now that we know k, we can solve for rate when time is equal to 2/45 hours.
formula becomes rate = 3.75 / (2/45).
this is the same as rate = 3.75 * 45 / 2 which results in rate = 84.375 miles per hour.
if he travels at 150 miles per hour, he covers 3.75 miles in one and a half minutes.
if he travels at 84.375 miles per hour, he covers 3.75 miles in two and two third minutes.
note that this problem could have been solved without doing the conversion from minutes to hours, but k would not then have represented the distance.
it would, however, still have been the constant of variation and could still be used to solve the problem.
in general, the inverse variation formula is y = k / x.
note that 1 and 1/2 minutes = 3/2 minutes and 2 and 2/3 minutes = 8/3 minutes.
if we let y = 150 and x = 3/2, then the formula becomes 150 = k / (3/2).
solve for k to get k = 225.
when k = 225 and x = 8/3, the formula becomes y = 225 / (8/3) which is the same as 225 * 3/8 which is equal to 84.375.
we got the same answer, even though we did not convert minutes to hours.
in this case, k was just the constant of variation and did not represent distance in miles.
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