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To find the present value of $5000 payable at the end of 3 years, with continuous compounding at an interest rate of 9%, we use the present value formula for continuously compounded interest:

\[
PV = FV \times e^{-rt}
\]

where:
- $ PV $ is the present value (what we are solving for),
- $ FV = 5000 $ is the future value (amount payable at the end of 3 years),
- $ r = 0.09 $ is the interest rate (9%),
- $ t = 3 $ is the time in years,
- $ e $ is the base of the natural logarithm (approximately 2.71828).

### Calculation:

\[
PV = 5000 \times e^{-(0.09)(3)}
\]

Let's calculate this:

1. Calculate the exponent: $ -(0.09)(3) = -0.27 $.
2. Find $ e^{-0.27} $.
3. Multiply the result by 5000 to get the present value.

Performing these calculations:

\[
PV \approx 5000 \times 0.7634 \approx 3817
\]

### Answer:
The present value of $5000 payable at the end of 3 years is approximately **$3817**.

Would you like more details or have any questions?

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Here are some related questions to expand on this topic:
1. What is the difference between simple interest and continuously compounded interest?
2. How would the present value change if the interest rate were increased to 10%?
3. How does continuous compounding compare to annual or monthly compounding for the same interest rate?
4. Can continuous compounding be applied to real-life financial products?
5. How would you adjust the formula if the interest rate changes during the 3-year period?

**Tip:** Continuous compounding maximizes the effect of interest over time, making it useful for calculating present values with precision.

Find the present value of $5000 payable at the end of 3 years, if money may be invested at 9% with interest compounded continuously.

Discover how to calculate the present value of $5000, payable in 3 years, using continuous compounding at a 9% interest rate. This example explores present value calculations and exponential functions, providing a detailed step-by-step solution for Grades 11-12.

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