Question

A recent report by the U.S Census Bureau predicts that the U.S Hispanic population will increase from 35.6 million in 2000 to 102.6 million in 2050. Assuming the exponential growth model fits this population growth, express the population P as a function of the year t. Let 2000 correspond to t = 0, write an initial value problem.

Answer 1

`y(t)=2000e^(kt)`

Explanation:

We are given that

`(dy)/(dt)=ky`

`y(0)=2000`

`\int(dy)/(y)=\int kdt`

`ln y=kt+c`

Using the formula

`\int(dx)/(x)=ln x`

`y=e^(kt+c)=e^(kt)\cdot e^c=Ce^(kt)`

Where
`C=e^c`

Substitute the values

`2000=C`

`y=2000e^(kt)`