Question

QL An initial investment amount P, an annual interest rate r, and a time t are given. Find the future value of the investment when interest is compounde monthly, (c) daily, and (d) continuously. Then find (e) the doubling time I for the given interest rate. P = $2500, r=3.95%, t = 8 yr Te luture value ore vesmen we erst is Compouleu annually iS J400.29 (Type an integer or a decimal. Round to the nearest cent as needed.) b) The future value of the investment when interest is compounded monthly is $3,427.30 (Type an integer or a decimal. Round to the nearest cent as needed.) c) The future value of the investment when interest is compounded daily is $3429.02 (Type an integer or a decimal. Round to the nearest cent as needed.) d) The future value of the investment when interest is compounded continuously is $ 3429.08 (Type an integer or a decimal. Round to the nearest cent as needed.) e) Find the doubling time for the given interest rate. T= yr (Type an integer or decimal rounded to two decimal noon

Answer 1

User asks to solve part e of the problem.

We are asked to find the doubling time for the interest rate in the case of continuous compounding.

Recall that the formula for the continuously compounding interest rate is:

`A=P\cdot e^((r\cdot t\)}`

Then, we need to solve when the value "A" (Accrued value) doubles the principal P, that is:

`2P=P\cdot e^((r\cdot t\)}`

Dividing both sides by P and then applying natural logarithms (ln) on both sides, we get:

`\begin{gathered} 2P=P\cdot e^((r\cdot t\)} \\ 2=e^((r\cdot t)) \\ \ln (2)=r\cdot t \\ \ln (2)=0.0395\cdot t \\ t=(\ln l(2))/(0.0395) \\ t=17.548 \end{gathered}`

which gives us approximately 17.548 years which we round to two decimals as: 17.55 years