Question

The formula A=23.1e0.0152t models the population of a US state, A, in millions, t years after 2000. Determine algebraically when the population was predicted to reach 28.3 million.

Answer 1

The given equation is

`A=23.1e^(0.0152t)`

Where A is the population from 2000 in t years

since the population in t years is 28.3 million, then

Substitute A by 28.3 to find t

`28.3=23.1e^(0.0152t)`

Divide both sides by 23.1

`\begin{gathered} (28.3)/(23.1)=(23.1)/(23.1)e^(0.0152t) \\ (283)/(231)=e^(0.0152t) \end{gathered}`

Insert ln for both sides

`\ln ((283)/(231))=\ln (e^(0.0152t))`

Use the rule

`\ln (e^n)=n`
`\ln ((283)/(231))=0.0152t`

Divide both sides by 0.0152 to find t

`\begin{gathered} (\ln ((283)/(231)))/(0.0152)=(0.0152t)/(0.0152) \\ 13.35718336=t \end{gathered}`

Round it to the nearest year, then

t = 14

Add 14 to 2000 to find the year

2000 + 14 = 2014, then

The population was 28.3 million at 2014