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The problem asks to use the continuous compound interest formula to find the time $ t $ when given the final amount $ A $, principal $ P $, and interest rate $ r $.

The formula for continuous compound interest is:

\[
A = P e^{rt}
\]

where:
- $ A $ is the final amount,
- $ P $ is the initial principal,
- $ r $ is the interest rate (as a decimal),
- $ t $ is the time in years,
- $ e $ is the base of the natural logarithm (approximately 2.71828).

We are given:
- $ A = 71,000 $,
- $ P = 47,143 $,
- $ r = 9.1\% = 0.091 $.

Now, we need to solve for $ t $. Follow these steps:

### Step 1: Plug in the values
\[
71,000 = 47,143 \cdot e^{0.091t}
\]

### Step 2: Divide both sides by $ 47,143 $
\[
\frac{71,000}{47,143} = e^{0.091t}
\]
\[
1.5067 \approx e^{0.091t}
\]

### Step 3: Take the natural logarithm of both sides
\[
\ln(1.5067) = 0.091t
\]
\[
0.4107 \approx 0.091t
\]

### Step 4: Solve for $ t $
\[
t = \frac{0.4107}{0.091} \approx 4.51 \, \text{years}
\]

Thus, $ t \approx 4.51 $ years.

Would you like more details on any of the steps?

Here are 5 related questions to explore:
1. How does continuous compound interest differ from regular compound interest?
2. Can you solve for $ r $ if $ t $ is known?
3. How would the formula change if interest were compounded monthly instead of continuously?
4. What is the effect of doubling the principal $ P $ on the time $ t $?
5. How does the value of $ e $ influence the behavior of the equation?

**Tip:** Always remember to convert percentage rates to decimals (e.g., 9.1% = 0.091) before using them in formulas.

Use the continuous compound interest formula to find the indicated value. A = $71,000; P = $47,143; r = 9.1%; t = ?

Learn how to use the continuous compound interest formula to solve for time. In this example, A = $71,000, P = $47,143, r = 9.1%, and t is unknown. Follow a detailed step-by-step solution involving natural logarithms and exponential functions. Suitable for Grades 10-12.

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