User x=1+r , where r is the interest rate paid each year. Write a model polynomial, C(x) . Represent the final amount of a 7-year investment if $5,000 was deposited at the beginning of the first year, and $2,000 was deposited at the beginning of the third year. (1 point)

Question

Responses

C(x)=5,000x+2,000x3
cap c times x is equal to 5 comma 000 x plus 2 comma 000 x cubed

C(x)=5,000x7+2,000x5
cap c times x is equal to 5 comma 000 x to the 7th power plus 2 comma 000 x to the 5th power

C(x)=5,000x7+7,000x4
cap c times x is equal to 5 comma 000 x to the 7th power plus 7 comma 000 x to the 4th power

C(x)=5,000x7+2,000x3

Answer 1

To model the final amount of the investment using a polynomial \(C(x)\), we need to account for the two deposits and how they grow over time at the interest rate \(r\) (which translates to \(x = 1 + r\)).

1. First Deposit: $5,000 deposited at the beginning of the first year. This deposit will grow for 7 years: \[ 5,000 \cdot x^7 \]

2. Second Deposit: $2,000 deposited at the beginning of the third year. This deposit will grow for 5 years (since it is deposited at the start of year 3 and will grow until the end of year 7): \[ 2,000 \cdot x^5 \]

Putting this together, the polynomial representing the total amount after 7 years is: \[ C(x) = 5,000x^7 + 2,000x^5 \]

Therefore, the correct response is:

C(x) = 5,000x^7 + 2,000x^5