(a)
(b)
(a)
(b)
The bacteria in a certain culture increase continuously at a rate proportional to
the number present.
If the number triples in 6 hours, how many will there be in 12 hours?
In how many hours will the original number quadruple?
SOLUTIONS:
We let N be the number at time t and N 0 the number initially. Then
hence, C = ln N 0 . The general solution is then N = N 0 ekt, with k still to be
determined.
Since N = 3N 0 when t = 6, we see that 3N 0 = N 0 e^6 k and that ln 3. Thus
N = N 0 e(t ln 3)/6.
When t = 12, N = N 0 e 2 ln 3 = N 0 eln 32 = N 0 eln 9 = 9N 0.
We let N = 4N 0 in the centered equation above, and getExample 14 __
Radium-226 decays at a rate proportional to the quantity present. Its half-life is
1612 years. How long will it take for one quarter of a given quantity of radium-
226 to decay?
SOLUTION:
If Q(t) is the amount present at time t, then it satisfies the equation
Q(t) = Q 0 e kt, (1)where Q 0 is the initial amount and k is the (negative) factor of proportionality.
Since it is given that when t = 1612, equation (1) yields