Answer
The bacteria population after three hours is $~11,713$
Work Step by Step
We can find the bacteria population after three hours:
$P(3) = 400+ \int_{0}^{3}r(t)~dt = 400+\int_{0}^{3}450.268~e^{1.12567~t}~dt$
Let $u = 1.12567~t$
$\frac{du}{dt} = 1.12567$
$dt = \frac{du}{1.12567}$
When $t = 0$, then $u = 0$
When $t= 3$, then $u = 3.37701$
$P(3) = 400+\int_{0}^{3.37701}(450.268~e^u)~(\frac{du}{1.12567})$
$P(3) = 400+\int_{0}^{3.37701}400~e^u~du$
$P(3) = 400+400~e^u\vert_{0}^{3.37701}$
$P(3) = 400+400~(e^{3.37701}-e^0)$
$P(3) = 400+400~(e^{3.37701}-1)$
$P(3) = 400+400~(28.283)$
$P(3) = 11,713$
The bacteria population after three hours is $~11,713$
