Answer
The number of bacteria after $t$ hours is: $~~n = (400)~(3^t)$
The rate of growth after $2.5~h$ is: $~~6850$ bacteria/hour
Work Step by Step
At $t = 0$:
$n_0 = 400$
At $t = 1~h$:
$n = (400)(3) = 1200$
At $t = 2~h$:
$n = (1200)(3) = (400)(3^2) = 3600$
At time $t$:
$n = n_0~3^t = (400)~(3^t)$
We can find the rate of growth after $2.5~h$:
$n = (400)~(3^t)$
$\frac{dn}{dt} = (400)~(3^t)(ln~3)$
$\frac{dn}{dt} = (400)~(3^{2.5})(ln~3)$
$\frac{dn}{dt} = 6850$ bacteria/hour
