Question

the population p, in thousands, of a small city is given by the following function, where t is time in years. answer parts a) through c) p(t)=\frac{400t}{3t^{2}+25} a) find the growth rate the growth rate is

Explanation:

Step1: Apply quotient - rule for differentiation

The quotient - rule states that if $y=\frac{u}{v}$, then $y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$. Here, $u = 400t$, $u^\prime=400$, $v = 3t^{2}+25$, $v^\prime = 6t$.

Step2: Calculate the derivative

$P^\prime(t)=\frac{400(3t^{2}+25)-400t\times6t}{(3t^{2}+25)^{2}}=\frac{1200t^{2}+10000 - 2400t^{2}}{(3t^{2}+25)^{2}}=\frac{10000 - 1200t^{2}}{(3t^{2}+25)^{2}}$

Answer:

$\frac{10000 - 1200t^{2}}{(3t^{2}+25)^{2}}$