(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. \( \qquad \) 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 an investment compounded daily, we 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. Given: - \( P = 3900 \) - \( r = 7.77\% = 0.0777 \) - \( n = 365 \) (compounded daily) - \( t = 22 \) Now, we can substitute these values into the formula: \[ A = 3900 \left(1 + \frac{0.0777}{365}\right)^{365 \times 22} \] Let's calculate the expression step by step. 1. Calculate \( \frac{0.0777}{365} \). 2. Calculate \( 365 \times 22 \). 3. Substitute these values into the formula and calculate \( A \). I'll perform these calculations now. Calculate the value by following steps: - step0: Calculate: \(365\times 22\) - step1: Multiply the numbers: \(8030\) Calculate or simplify the expression \( 0.0777/365 \). Calculate the value by following steps: - step0: Calculate: \(\frac{0.0777}{365}\) - step1: Convert the expressions: \(\frac{\frac{777}{10000}}{365}\) - step2: Multiply by the reciprocal: \(\frac{777}{10000}\times \frac{1}{365}\) - step3: Multiply the fractions: \(\frac{777}{10000\times 365}\) - step4: Multiply: \(\frac{777}{3650000}\) Calculate or simplify the expression \( 3900*(1+0.000213)^(8030) \). Calculate the value by following steps: - step0: Calculate: \(3900\left(1+0.000213\right)^{8030}\) - step1: Add the numbers: \(3900\times 1.000213^{8030}\) - step2: Convert the expressions: \(3900\left(\frac{1000213}{1000000}\right)^{8030}\) - step3: Simplify: \(3900\times \frac{1000213^{8030}}{1000000^{8030}}\) - step4: Rewrite the expression: \(100\times 39\times \frac{1000213^{8030}}{1000000^{8030}}\) - step5: Rewrite the expression: \(100\times 39\times \frac{1000213^{8030}}{100^{24090}}\) - step6: Reduce the numbers: \(39\times \frac{1000213^{8030}}{100^{24089}}\) - step7: Multiply: \(\frac{39\times 1000213^{8030}}{100^{24089}}\) Now that we have the necessary calculations: 1. The value of \( \frac{0.0777}{365} \) is approximately \( 0.000213 \). 2. The value of \( 365 \times 22 \) is \( 8030 \). Substituting these values into the compound interest formula: \[ A = 3900 \left(1 + 0.000213\right)^{8030} \] Calculating this gives us: \[ A \approx 21567.43 \] Thus, the value in year \( x=22 \) of an investment of \( \$ 3,900 \) dollars which pays \( 7.77\% \) compounded daily is approximately: \[ \$ 21567.43 \] So, the final answer rounded to the nearest cent is: \[ \boxed{21567.43} \]