Questions: Suppose that 25,000 from a retirement account is invested in a large cap stock fund. After 30 yr, the value is 177,856.06. Part 1 of 2 (a) Use the model A=P e^rt to determine the average rate of return under continuous compounding: Round to the nearest tenth of a percent. Avoid rounding in intermediate steps. The average rate is approximately %.

Suppose that 25,000 from a retirement account is invested in a large cap stock fund. After 30 yr, the value is 177,856.06. Part 1 of 2 (a) Use the model A=P e^rt to determine the average rate of return under continuous compounding: Round to the nearest tenth of a percent. Avoid rounding in intermediate steps. The average rate is approximately %.
Transcript text: Suppose that $\$ 25,000$ from a retirement account is invested in a large cap stock fund. After 30 yr , the value is $\$ 177,856.06$. Part 1 of 2 (a) Use the model $A=P e^{r t}$ to determine the average rate of return under continuous compounding: Round to the nearest tenth of a percent. Avoid rounding in intermediate steps. The average rate is approximately $\square$ $\%$.
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Solution

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Solution Steps

Step 1: Start with the continuous compounding formula: \(A = Pe^{rt}\).
Step 2: Rearrange the formula to solve for \(r\): \(r = \frac{1}{t} \ln\left(\frac{A}{P}\right)\).
Step 3: Substitute the given values into the formula: \(r = \frac1{30} \ln\left(\frac{177856.06}{25000}\right)\).
Step 4: Calculate \(r\) using the values substituted in step 3: \(r = 0.0654\).
Step 5: Convert \(r\) into a percentage: \(r = 6.540\%\).
Step 6: Round \(r\) to the nearest tenth of a percent, as specified: \(r = 6.5\%\).

Final Answer: The average rate of return under continuous compounding is approximately 6.5%.

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