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 \( r \).