To model the final amount of a 7-year investment with two deposits ($5,000 at the beginning of the first year and $2,000 at the beginning of the third year) based on the interest rate \( r \), we will use the variable \( x = 1 + r \).
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First Deposit: For the $5,000 deposited at the beginning of the first year, it will earn interest for the full 7 years. The future value of this deposit can be represented as: \[ 5,000 \cdot x^7 \]
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Second Deposit: For the $2,000 deposited at the beginning of the third year, it will earn interest for 5 years (since it is deposited at the beginning of year 3 and we are looking at a total of 7 years). The future value of this deposit can be represented as: \[ 2,000 \cdot x^5 \]
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Combine both deposits: The total future value, \( C(x) \), can be represented by adding the future values of both deposits: \[ C(x) = 5,000 \cdot x^7 + 2,000 \cdot x^5 \]
Thus, the correct polynomial model for the final amount of the investment after 7 years is: \[ \boxed{C(x) = 5,000x^7 + 2,000x^5} \]