Question

The Johnson family wants to start a college fund for their daughter Gabriella. They put $63, 000 into an account that grows at a rate of 2.55% per year, compounded quarterly. Given the function G(t)= 63,000(1+ .0255/4)^ 4t, where G(t) represents the amount of money in the account t years after the account is opened, given that no more money is deposited into or withdrawn from the account.

Calculate the number of years it will take for the account to reach approximately $150,000, to the nearest hundredth of a year.

Answer 1

It will take approximately 34.13 years

Explanation:

The function G(t) below represents the amount of money in some account t years after the account is opened for The Johnson's daughter Gabriella:

`G(t)= 63,000(1+ .0255/4)^ {4t}`

It's required to find the number of years (t) it will take for the account to reach G(t)=150,000. We need to solve the equation:

`63,000(1+ .0255/4)^ {4t}=150,000`

Dividing by 63,000 and simplifying:

`\displaystyle (1+ .0255/4)^ {4t}=(150,000)/(63,000)=2.38095`

Taking logarithms:

`\displaystyle \log(1+ .0255/4)^ {4t}=\log 2.38095`

Applying logarithms property:

`\displaystyle (4t) \log(1+ .0255/4)=\log 2.38095`

Solving for t:

`\displaystyle 4t =(\log 2.38095)/(\log(1+ .0255/4))`

`\displaystyle t =(\log 2.38095)/(4\log(1+ .0255/4))`

Calculating:

`\displaystyle t =(0.37675)/(0.01104)`

`\boxed{t \approx 34.13}`

It will take approximately 34.13 years