Question

At what interest compounded continuously must you invest $4900 to have$10740.65 money in 9 years? First round the interest rate r to 4 decimalplaces and then re-write it as aa percentage with 2 decimal places.

Answer 1

`\begin{gathered} r=0.0872 \\ r=8.72\% \end{gathered}`

Step-by-step explanation

We want to find the rate at which the amount was continuously compounded.

The formula for the total amount for a continuously compounded principal is:

`A=Pe^(rt)`

where A = amount

P = principal

r = rate

t = time (in years)

Substituting the given values into the equation:

`\begin{gathered} 10740.65=4900\cdot e^(r\cdot9) \\ 10740.65=4900\cdot e^(9r) \end{gathered}`

Divide both sides by 4900:

`\begin{gathered} (10740.65)/(4900)=e^(9r) \\ e^(9r)=2.1920 \end{gathered}`

Find the natural logarithm of both sides of the equation:

`\begin{gathered} \ln (e^(9r))=\ln 2.1920 \\ 9r=0.7848 \end{gathered}`

Divide both sides by 9:

`\begin{gathered} r=(0.7848)/(9) \\ r=0.0872 \end{gathered}`

Convert to decimal number:

`\begin{gathered} r=0.0872\cdot100 \\ r=8.72\% \end{gathered}`

That is the interest rate.