Question

I need help with this homework question please and thankyou

Answer 1

The formula for continuously compounded interest is

`\begin{gathered} A=Pe^(rt) \\ \text{ Where }A\text{ is the Amount or future value} \\ P\text{ is the Principal, or initial value} \\ r\text{ is the interest rate, and} \\ t\text{ is the time} \end{gathered}`

So, in this case, we have

`\begin{gathered} A=\text{ \$}1,000 \\ P=\text{ \$}200 \\ r=4\text{\% }=(4)/(100)=0.04 \\ t=\text{ ?} \end{gathered}`
`\begin{gathered} A=Pe^(rt) \\ \text{ Replace the know values} \\ \text{\$}1,000=\text{\$}200\cdot e^(0.04t) \\ \text{ Divide by \$200 from both sides of the equation} \\ \frac{\text{\$}1,000}{\text{\$}200}=\frac{\text{\$}200\cdot e^(0.04t)}{\text{\$}200} \\ 5=e^(0.04t) \\ \text{ Apply natural logarithm to both sides of the equation} \\ \ln (5)=\ln (e^(0.04t)) \\ \ln (5)=0.04t \\ \text{ Divide by 0.04 from both sides of the equation} \\ (\ln(5))/(0.04)=(0.04t)/(0.04) \\ \boldsymbol{40.2\approx t} \end{gathered}`

Therefore, it will take approximately 40 years for the account to reach $1,000.