SOLUTION: A rumor at Hearsay High School is started by 2 students and grows exponentially. 12 minutes later, 15 students know the rumor. How long will it take for the entire population of He

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Question 1129160: A rumor at Hearsay High School is started by 2 students and grows exponentially. 12 minutes later, 15 students know the rumor. How long will it take for the entire population of Hearsay (1,250 students) to hear the rumor (nearest minute).
Answer by greenestamps(13215) About Me  (Show Source):
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In 12 minutes, the number of students who know the rumor increases from 2 to 15, a growth factor of 15/2 = 7.5. The number will continue growing by a factor of 7.5 every 12 minutes; we want to know how many minutes it will take for all 1250 students to know the rumor.

If n is the number of 12-minute intervals, then the number of students who know the rumor is given by the exponential function

2%287.5%29%5En

So we need to solve

2%287.5%29%5En+=+1250 or %287.5%29%5En+=+625

With the n as an exponent, we need to use logarithms:

log%28%28%287.5%29%5En%29%29+=+log%28%28625%29%29
n%2Alog%28%287.5%29%29+=+log%28%28625%29%29
n+=+log%28%28625%29%29%2Flog%28%287.5%29%29 = 3.195 to 3 decimal places.

So the time required for all 1250 students to know the rumor is 12*3.195 = 38.34; or 38 minutes to the nearest minute.