Answer
The rate at which the population of bacteria is growing when $t = 2$ is $31.5501$ bacteria/hour.
Work Step by Step
Rewrite $P(t)$:
$P(t) = 500 + \frac{2000t}{50 + t^{2}}$
Find the derivative of $P(t)$ using the Quotient Rule:
$P'(t) = \frac{(2000)(50 + t^2) - (2000t)(2t)}{(50 + t^{2})^{2}}$
Find $P'(2)$ to find the rate when $t = 2$:
$P'(2) = \frac{(2000)(50 + (2)^2) - (2000(2))(2(2))}{(50 + (2)^{2})^{2}}$
$= 31.5501$ bacteria/hour
