Question

It is predicted that in t years, the population of a country will be P (t) = 50e ^ (0.02t) million inhabitants. A) What will be the rate of change of the population in 10 years. B) What will be the relative rate of the population in t years? Is this rate constant?

Answer 1

`\begin{gathered} \text{Given:} \\ P(t)=50e^(0.02t) \end{gathered}`

A) What will be the rate of change of the population in 10 years.

`\begin{gathered} \text{Get the derivative of the }P(t)\text{ with respect to }t \\ (dP)/(dt)(50e^(0.02t))=50(0.02)e^(0.02t) \\ (dP)/(dt)(50e^(0.02t))=e^(0.02t) \end{gathered}`

Substitute t = 10, to the derivative of P(t)

`\begin{gathered} P^(\prime)(t)=e^(0.02t) \\ P^(\prime)(10)=e^(0.02(10)) \\ P^(\prime)(10)=e^(0.2) \\ P^(\prime)(10)=e^(0.2) \\ P^(\prime)(10)=1.02020134 \\ \\ \text{Round off to two decimal place} \\ P^(\prime)(10)=1.02 \end{gathered}`

Therefore, the rate of change of the population in 10 years is 1.02 million.

B) What will be the relative rate of the population in t years? Is this rate constant?​

`\begin{gathered} \text{The relative rate of the population in t years is the first derivative of }P(t) \\ P^(\prime)(t)=e^(0.02t) \end{gathered}`

The rate is not constant, as it depends on how much time t has passed.