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)%0D%0A𝐶%0D%0A(%0D%0A𝑥%0D%0A)%0D%0A, 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)%0D%0AResponses

Answer 1

To model the final amount of the investment over 8 years with the described deposits and compounding interest, we can use the expression for the compounded amount for each deposit made.

1. For the first deposit of $500 at the beginning of year 1 (which will earn interest for 8 years): \[ A_1 = 500 \cdot (1 + r)^8 \]

2. For the second deposit of $800 at the beginning of year 4 (which will earn interest for 5 years): \[ A_2 = 800 \cdot (1 + r)^5 \]

3. For the third deposit of $370 at the beginning of year 8 (which will earn interest for 0 years – no interest accumulates since it's made at the end): \[ A_3 = 370 \cdot (1 + r)^0 = 370 \]

Now we can combine these amounts to find the total amount \( C(x) \) after 8 years: \[ C(x) = A_1 + A_2 + A_3 \] \[ C(x) = 500(1 + r)^8 + 800(1 + r)^5 + 370 \]

This polynomial \( C(x) \) represents the final amount of the investment after 8 years, taking into account the different amounts deposited and the interest rate \( r \).