Question

The formula Upper A equals 23.1 e Superscript 0.0152 tA=23.1e0.0152t models the population of a US​ state, A, in​ millions, t years after 2000.

a. What was the population of the state in​ 2000?
b. When will the population of the state reach 28.328.3 ​million?

Answer 1

a)
`A(t=0)= 23.1 e^(0.0152(0))=23.1e^0 =23.1`

b)
`t = (ln((28.3)/(23.1)))/(0.0152)=13.357 years`

So for this case the answer would be 13.347 years, the population will be 28.3 million and ye year would be 2000+13.347 and that would be approximately in 2014

Explanation:

For this case we assume the following model:

`A(t)= 23.1 e^(0.0152 t)`

Where t is the number of years after 2000/

Part a

For this case we want the population for 2000 and on this case the value of t=0 since we have 0 years after 2000. If we rpelace into the model we got:

`A(t=0)= 23.1 e^(0.0152(0))=23.1e^0 =23.1`

So then the initial population at year 2000 is 23.1 million of people.

Part b

For this case we want to find the time t whn the population is 28.3 million.

So we need to solve this equation:

`28.3= 23.1 e^(0.0152(t))`

We can divide both sides by 23.1 and we got:

`(28.3)/(23.1)= e^(0.0152t)`

Now we can apply natural log on both sides and we got:

`ln((28.3)/(23.1))= 0.0152 t`

And then for t we got:

`t = (ln((28.3)/(23.1)))/(0.0152)=13.357 years`

So for this case the answer would be 13.347 years, the population will be 28.3 million and ye year would be 2000+13.347 and that would be approximately in 2014