Answer to Question #5211 in Differential Equations for Elsie

Question

Bacteria in a petrie dish double the area they cover everyday. If the dish is covered after 16 days, on what day was only one quarter of it covered?

Answer 1

Let the area of the whole dish be 1.
Suppose the bacteria covered an area S
at the beginning.
After the first day it covered the area 2S, after the
second - 4S and so on.
So, after n days the bacteria covered (2^n)*S. We know
that after 16 days it covered the dish. We get:

(2^16)*S = 1, S =
1/(2^16) = 2^(-16).

Let x be the number of days that after x days only
one quarter was covered. Then

(2^x) * S = 1/4,
(2^x) * 2^(-16) =
2^(-2),
2^(x-16) = 2^(-2),
x - 16 = -2,
x = 14.