Suppose \( \$ 3,900 \) is deposited into an account which has an annual percentage rate of \( 7.77 \% \) per year. Assume it remains in the account for 22 years and no additional money is added to the account other than interest. (a) Complete the boxes below to write an expression for the amount (in dollars) at year \( x=22 \), of assuming interest is compounded annually. Do not round any values. You can enter arithmetic expressions (containing \( +,-, *, / \), or \( \wedge \) ) in any of these boxes. Number Number \( \cdot( \) Number ) What is the value in year \( x=22 \) of an investment of \( \$ 3,900 \) dollars which pays \( 7.77 \% \) compounded annually? \$ \( \square \) Number (Round to the nearest 0.01 dollars)

To find the amount in the account at year \( x = 22 \) with an initial deposit of \( \$3,900 \) and an annual interest rate of \( 7.77\% \) compounded annually, we can use the compound interest formula: \[ A = P(1 + r)^t \] where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (\$3,900). - \( r \) is the annual interest rate (decimal) (7.77% = 0.0777). - \( t \) is the time the money is invested for in years (22). (a) The expression for the amount at year \( x = 22 \) is: \[ \$3,900 \cdot (1 + 0.0777)^{22} \] Now, calculating the value: 1. Calculate \( (1 + 0.0777)^{22} \): \[ (1.0777)^{22} \approx 4.6884231 \] 2. Now multiply by the principal: \[ A \approx 3,900 \cdot 4.6884231 \approx 18,284.0261 \] Rounding to the nearest 0.01 dollars, the value in year \( x = 22 \) would be: \[ \$ 18,284.03 \] So, enter the calculations accordingly and the final rounded value is \$ \( 18,284.03 \).