Question

A particular​ country's exports of goods are increasing exponentially. The value of the​ exports, t years after 2008​, can be approximated by ​V(t)equals1.4 e Superscript 0.039 t where tequals0 corresponds to 2008 and V is in billions of dollars.

Answer 1

a) For 2008 we have that t = 2008-2008 = 0 and we have:

`V(0)= 1.4e^(0.039*0)= 1.4`

For 2022 we have that t = 2022-2008=14 and if we replace we got:

`V(12) = 1.4 e^(0.039*14)=2.417`

b)
`2.8 = 1.4 e^(0.039 t)`

We can divide both sides by 1.4 and we got:

`2 = e^(0.039 t)`

Now natural log on both sides:

`ln (2) = 0.039 t`

`t = (ln(2))/(0.039)= 17.77 years`

Explanation:

For this case we have the following model given:

`V(t) = 1.4 e^(0.039 t)`

Where V represent the exports of goods and the the number of years after 2008.

Part a

Estimate the value of the country's exports in 2008 and 2022

For 2008 we have that t = 2008-2008 = 0 and we have:

`V(0)= 1.4e^(0.039*0)= 1.4`

For 2022 we have that t = 2022-2008=14 and if we replace we got:

`V(12) = 1.4 e^(0.039*14)=2.417`

Part b

What is the doubling time for the value of the country's exports.

For this case we can set up the following condition:

`2.8 = 1.4 e^(0.039 t)`

We can divide both sides by 1.4 and we got:

`2 = e^(0.039 t)`

Now natural log on both sides:

`ln (2) = 0.039 t`

`t = (ln(2))/(0.039)= 17.77 years`