(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. \( \qquad \) 3900 \( \cdot(1+0.0777 / 365) \) 8030 What is the value in year \( x=22 \) of an investment of \( \$ 3,900 \) dollars which pays \( 7.77 \% \) compounded daily? \$ \( \square \) 21,547.50 (Round to the nearest 0.01 dollars)

To calculate the future value of the investment after 22 years with daily compounding interest, you can use the formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial amount of money, which is \$3,900). - \( r \) is the annual interest rate (decimal) (7.77% = 0.0777). - \( n \) is the number of times that interest is compounded per year (365). - \( t \) is the number of years the money is invested for (22). Plugging in the values: \[ A = 3900 \left(1 + \frac{0.0777}{365}\right)^{365 \cdot 22} \] Calculating step by step: 1. Calculate \( \frac{0.0777}{365} \approx 0.0002128767 \). 2. Calculate \( 365 \cdot 22 = 8030 \). 3. Now calculate \( 1 + 0.0002128767 \approx 1.0002128767 \). 4. Raise it to the power of 8030: \( (1.0002128767)^{8030} \). This will give you approximately \( 6.85187 \) as the result. Calculate: \[ A \approx 3900 \cdot 6.85187 \approx 26,476.24 \] Therefore, the future value of the investment after 22 years, rounded to the nearest cent, is: \$ \( 26,476.24 \) Make sure to double-check calculations when mixing rounding and arithmetic—every tiny digit counts in finance! You can explore further by practicing on different interest rates or compounding frequencies to see how it impacts your returns. And if you want to dive deeper, there are plenty of resources out there on finance basics and the intricacies of exponential growth in investments!