Question

The size P of a small herbivore population at time t (in years) obeys the function P(t) = 1000e0-16t if they have enough food and the predator population stays constant. After how many years will the population reach 4000?

Answer 1

After 9 years, the population reach 4000.

Explanation:

The given function P(t) represents the size of a small herbivore population at time t (in years).

`P(t)=1000e^(0.16t)`

We need to find the number of years after which population reach 4000.

Substitute P(t)=4000 in the above function.

`4000=1000e^(0.16t)`

Divide both sides by 1000.

`4=e^(0.16t)`

Taking ln on both sides.

`\ln 4=\ln e^(0.16t)`

`1.3863=0.16t`
`[\because \ln e^x=x]`

Divide both sides by 0.16.

`(1.3863)/(0.16)=t`

`8.664375=t`

We need to find the number of years. So, round the answer to the next whole number.

`t\approx 9`

Therefore, after 9 years, the population reach 4000.