To help pay for new costumes for a​ play, a theater invests ​$1300 in an 18​-month CD paying 4.8​% interest compounded monthly. Determine the amount the theater will receive when it cashes in the CD after 18 months.
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Part 1
The theater will receive ​$
  
enter your response here when it cashes in the CD.
​(Round to the nearest cent as​ needed.)

To solve this problem, we can use the compound interest formula:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

where:
- $ A $ is the amount of money the theater will receive after the investment period.
- $ P $ is the principal, or the initial amount invested, which is $1300.
- $ r $ is the annual interest rate (in decimal form), which is 4.8%, or 0.048.
- $ n $ is the number of times interest is compounded per year, which is monthly (12 times per year).
- $ t $ is the time in years, which is 18 months or $ \frac{18}{12} = 1.5 $ years.

Substitute these values into the formula:

\[
A = 1300 \left(1 + \frac{0.048}{12}\right)^{12 \times 1.5}
\]

Now, let’s calculate this step by step:

1. Calculate the monthly interest rate:
   \[
   \frac{0.048}{12} = 0.004
   \]

2. Calculate the exponent:
   \[
   12 \times 1.5 = 18
   \]

3. Substitute and calculate the total amount:
   \[
   A = 1300 \times (1 + 0.004)^{18}
   \]
   \[
   A = 1300 \times (1.004)^{18}
   \]

4. Raising 1.004 to the power of 18:
   \[
   (1.004)^{18} \approx 1.07393
   \]

5. Final calculation:
   \[
   A \approx 1300 \times 1.07393 \approx 1396.11
   \]

So, the theater will receive approximately **$1396.11** when it cashes in the CD after 18 months.

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Would you like further details on any step or have additional questions?

Here are some related questions that build on this:

1. How does changing the compounding frequency affect the final amount?
2. What would the amount be if the investment period were 24 months?
3. How would the calculation change if the interest were compounded quarterly instead?
4. How much interest does the theater earn on this investment?
5. What if the interest rate were increased to 6%?

**Tip:** When calculating compound interest, always double-check the compounding frequency and time period, as they significantly impact the final amount.

To help pay for new costumes for a play, a theater invests $1300 in an 18-month CD paying 4.8% interest compounded monthly. Determine the amount the theater will receive when it cashes in the CD after 18 months.

Find the future value of a $1300 investment in an 18-month CD with 4.8% interest compounded monthly. This problem includes detailed steps using the compound interest formula, making it ideal for high school students in Grades 10-12.

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