Questions: The formula A=Pe^(rt) describes the accumulated value, A, of a sum of money, P, the principal, after t years at annual percentage rate r (in decimal form) compounded continuously. Complete the table for a savings account subject to continuous compounding. Amount Invested Annual Interest Rate Accumulated Amount Time t in Years 8000 10% Double the amount invested ? t ≈ years (Do not round until the final answer. Then round to one decimal place as needed.)

The formula A=Pe^(rt) describes the accumulated value, A, of a sum of money, P, the principal, after t years at annual percentage rate r (in decimal form) compounded continuously. Complete the table for a savings account subject to continuous compounding.

Amount Invested Annual Interest Rate Accumulated Amount Time t in Years
8000 10% Double the amount invested ?

t ≈ years
(Do not round until the final answer. Then round to one decimal place as needed.)

Transcript text: The formula $\mathrm{A}=\mathrm{Pe} e^{t}$ describes the accumulated value, A , of a sum of money, P , the principal, after t years at annual percentage rate r (in decimal form) compounded continuously. Complete the table for a savings account subject to continuous compounding. \begin{tabular}{|c|c|c|c|} \hline Amount Invested & Annual Interest Rate & Accumulated Amount & Time t in Years \\ \hline$\$ 8000$ & $10 \%$ & Double the amount invested & $?$ \\ \hline \end{tabular} $t \approx$ $\square$ years (Do not round until the final answer. Then round to one decimal place as needed.)

Solution

Solution Steps

Step 1: Divide the Accumulated Amount by the Principal Amount

To find the time it takes for the principal amount $P = 8000$ to grow to the accumulated amount $A = 16000$ at an annual interest rate $r = 0.1$ (in decimal form) compounded continuously, we first divide $A$ by $P$. \[\frac{A}{P} = \frac{16000}{8000} = 2\]

Step 2: Take the Natural Logarithm of the Result from Step 1

Next, we take the natural logarithm (ln) of the result from step 1 to isolate the term containing $t$. \[\ln\left(\frac{A}{P}\right) = \ln(2) = 0.693\]

Step 3: Divide the Result from Step 2 by the Annual Interest Rate to Find the Time $t$

Finally, we divide the result from step 2 by the annual interest rate $r$ to find the time $t$ in years. \[t = \frac{\ln\left(\frac{A}{P}\right)}{r} = \frac{0.693}{0.1} = 6.9\]