Question

Use x=1+r%0D%0A𝑥%0D%0A=%0D%0A1%0D%0A+%0D%0A𝑟%0D%0A, where r is the interest rate paid each year. Write a model polynomial, C(x), that represents the final amount of an 8-year investment if $500 was deposited at the beginning of the first year, $800 was deposited at the beginning of the fourth year, and an additional $370 was deposited at the beginning of the eighth year.(1 point)

Answer 1

To model the final amount of the investment after 8 years with different deposit amounts and an interest rate \( r \), we can break down each deposit and how it accrues interest over time.

1. **Initial Deposit of $500 at the beginning of Year 1**:
- This amount will accrue interest for 8 years.
- The future value of this deposit will be calculated as:
\[
FV_1 = 500(1 + r)^8
\]

2. **Deposit of $800 at the beginning of Year 4**:
- This amount will accrue interest for 5 years (from Year 4 to Year 8).
- The future value of this deposit will be:
\[
FV_2 = 800(1 + r)^5
\]

3. **Deposit of $370 at the beginning of Year 8**:
- This amount will not accrue any interest because it is deposited at the end of Year 8.
- Thus, the future value of this deposit is simply:
\[
FV_3 = 370
\]

### Total Future Value

Now, we can combine all these future values into a single expression for the total amount \( C(x) \):

\[
C(r) = FV_1 + FV_2 + FV_3
\]
\[
C(r) = 500(1 + r)^8 + 800(1 + r)^5 + 370
\]

This polynomial \( C(r) \) represents the final amount after 8 years based on the interest rate \( r \).