Answer to Question #349168 in Differential Equations for Un Bunthim

Question

There are 10000 people living in a certain city. Suppose that the rate of population growth in the city is proportional to the number of inhabitants. Suppose that 20% of the original amount increase in 20 years, how much will the population in the city after 40 years?

Answer 1

$$\dfrac{dy}{dt}=ky$$ $$y(t)=y(0)e^{kt}$$

Given $y(0)=10000, y(20)=1.2y(0).$

Then

$$y(20)=y(0)e^{20k}=1.2y(0)$$

$$e^{20k}=1.2$$

$$y(40)=y(0)e^{k(40)}$$

$$y(40)=y(0)(1.2)^2$$

$$y(40)=1.44y(0)$$

$$y(40)=1.44(10000)$$

$$y(40)=14400$$

There will be 14400 people in a city after 40 years.