Question

Determining the proportional rates of growth for the following function: 0.015531t t Pe Derive a general expression for the proportional rate of growth. (1 mark) Calculate the proportional rates of growth at t = 5.

Answer 1

Solution;

Given;

$P(t)=0.01553t+P_0e^t$

The proportional rate of growth is the derivative of the function;

$\frac{dP}{dt}=0.015531+P_0e^t$ ....(1)

When there is no growth the rate of growth is 0;

$\frac{dP}{dt}=0=0.015531+P_0$

$P_0=-0.015531$

Now equation (1) becomes;

$\frac{dP}{dt}=0.015531-0.015531e^t$

Now ,rate at t=5.

By direct substitution;

$\frac{dP}{dt}=0.015531-0.015531e^5$

$\frac{dP}{dt}=-2.2894$

(Negative rate shows that the population is decreasing)