Question

If the number of students at a particular High School who participate in after-school drama programs increases at a rate of 8% per year, how long will it take for the number of students participating in the after-school programs to double?

a. about 25 years
b. about 12.5 years
c. about 3.6 years
d. about 9 years​

Answer 1

d. about 9 years​

Explanation:

There is a "rule of thumb" for doubling time* that says the product of the percentage rate of change per year and the doubling time in years is about 72. Here, that means the doubling time is about ...

72/8 = 9 . . . . years

_____

You can write the exponential equation ...

multiplier = (1 +.08)^n

and solve for multiplier = 2:

2 = 1.08^n

log(2) = n·log(1.08) . . . . . take logs

log(2)/log(1.08) = n . . . . . divide by the coefficient of n

9.00647 ≈ n

It will take about 9 years for the participation to double.

_____

* The farther away from 8% the rate of change is, or the more times per year it is compounded, the less accurate is the "rule of 72." When compounding is continuous, the "rule of 72" becomes the "rule of 69.4". For this problem, answer choices are sufficiently far apart that the rule of thumb is adequate for making a correct choice.