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.
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)
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