Lisa McNeil needs to choose between two​ investments: One pays 9.8​% compounded continuously​, and the other pays 10​% compounded quarterly. If she plans to invest ​$10 comma 000 for 2 ​years, which investment should she​ choose? How much extra interest will she earn by making the better​ choice?
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Part 1
She should choose the 
▼ 
 She will earn ​$
  
enter your response here extra interest using this choice.
​(Round to the nearest cent as​ needed.)

To compare the two investment options, we need to calculate the future value of each investment and then compare the amount of interest earned for each.

### Investment 1: 9.8% compounded continuously
The formula for continuous compounding is:
\[
A = P \cdot e^{rt}
\]
Where:
- $ A $ is the future value of the investment
- $ P = 10,000 $ (the principal)
- $ r = 0.098 $ (the annual interest rate)
- $ t = 2 $ years
- $ e $ is the base of the natural logarithm (approximately 2.718)

### Investment 2: 10% compounded quarterly
The formula for quarterly compounding is:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- $ A $ is the future value of the investment
- $ P = 10,000 $ (the principal)
- $ r = 0.10 $ (the annual interest rate)
- $ n = 4 $ (since the interest is compounded quarterly)
- $ t = 2 $ years

Let’s calculate the future values for both investments and find out how much extra interest she would earn with the better option.- The future value of the first investment (9.8% compounded continuously) is **$12,165.27**.
- The future value of the second investment (10% compounded quarterly) is **$12,184.03**.

Thus, Lisa should choose the second investment (10% compounded quarterly), as it yields more. She will earn an extra **$18.76** by choosing this option (rounded to the nearest cent).

Would you like further details or have any questions?

Here are some related questions:
1. How does the frequency of compounding affect the total interest earned?
2. What would be the result if both investments were compounded annually?
3. What is the difference between continuous and discrete compounding methods?
4. How much would Lisa earn if she invested for 5 years instead of 2 years?
5. What if the interest rate for the continuously compounded investment was increased to 10%?

**Tip:** Continuous compounding generally yields slightly less than more frequent discrete compounding at the same rate, unless the interest rate for continuous compounding is significantly higher.

Lisa McNeil needs to choose between two​ investments: One pays 9.8​% compounded continuously​, and the other pays 10​% compounded quarterly. If she plans to invest $10,000 for 2 years, which investment should she​ choose? How much extra interest will she earn by making the better​ choice?

Learn how to compare two investment options: one paying 9.8% compounded continuously and the other 10% compounded quarterly. This example calculates future value for $10,000 invested for 2 years and determines the better investment option. Detailed step-by-step solution with extra interest calculations.

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