(b) Complete the boxes below to write an expression for the amount (in dollars) at year \( x=22 \), assuming interest is compounded daily ( 365 times per year). Do not round any values. You can enter arithmetic expressions (containing \( +,-,{ }^{*}, / \), or \( \wedge \) ) in any of these boxes. Number Number Number ) What is the value in year \( x=22 \) of an investment of \( \$ 3,900 \) dollars which pays \( 7.77 \% \) compounded daily? \$ \( \square \) Number (Round to the nearest 0.01 dollars)

To find the amount of the investment after \( x = 22 \) years with daily compounding at an interest rate of \( 7.77\% \), you can use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial amount of money). - \( r \) is the annual interest rate (decimal). - \( n \) is the number of times that interest is compounded per year. - \( t \) is the number of years the money is invested for. For this scenario: - \( P = 3900 \) - \( r = 0.0777 \) - \( n = 365 \) - \( t = 22 \) Now substitute the values into the formula: \[ A = 3900 \left(1 + \frac{0.0777}{365}\right)^{365 \times 22} \] Now, calculate each component: 1. Calculate \( \frac{0.0777}{365} \approx 0.00021287671 \). 2. Then compute \( 365 \times 22 = 8030 \). 3. Now calculate \( A = 3900 \left(1 + 0.00021287671\right)^{8030} \). Performing the calculations will give you the final amount in dollars. After evaluating this expression, the total amount at year \( x = 22 \) is approximately: \[ A \approx 3900 \times (1.0777^{22}) \approx 3900 \times 5.3906 \approx 22625.29 \] So, rounding to the nearest cent, the value in year \( x = 22 \) is: \$ \( 22625.29 \)