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Answer to Question #5211 in Differential Equations for Elsie

Question #5211
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?
Expert's answer
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.

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