Brandon is going to invest in an account paying an interest rate of \( 4.6 \% \) compounded annually. How much would Brandon need to invest, to the nearest ten dollars, for the value of the account to reach \( \$ 69,000 \) in 8 years?

To find the initial investment needed for the account to reach a value of $69,000 in 8 years with an annual interest rate of 4.6% compounded annually, we can use the formula for compound interest: \[ A = P \left( 1 + \frac{r}{100} \right)^n \] where: - \( A \) is the future value of the investment, - \( P \) is the principal amount (initial investment), - \( r \) is the annual interest rate, - \( n \) is the number of years. Given: - \( A = \$69,000 \), - \( r = 4.6\% \), - \( n = 8 \) years. We need to solve for \( P \) to find the initial investment needed. Solve the equation by following steps: - step0: Solve for \(P\): \(69000=P\left(1+\frac{4.6}{100}\right)^{8}\) - step1: Simplify: \(69000=\frac{523^{8}}{500^{8}}\times P\) - step2: Swap the sides: \(\frac{523^{8}}{500^{8}}\times P=69000\) - step3: Rewrite the expression: \(\frac{523^{8}P}{500^{8}}=69000\) - step4: Cross multiply: \(523^{8}P=500^{8}\times 69000\) - step5: Simplify the equation: \(523^{8}P=69000\times 500^{8}\) - step6: Divide both sides: \(\frac{523^{8}P}{523^{8}}=\frac{69000\times 500^{8}}{523^{8}}\) - step7: Divide the numbers: \(P=\frac{69000\times 500^{8}}{523^{8}}\) The initial investment needed for the account to reach a value of $69,000 in 8 years with an annual interest rate of 4.6% compounded annually is \( P = \frac{69000 \times 500^8}{523^8} \). To find the value of \( P \) to the nearest ten dollars, we can calculate it. Calculate the value by following steps: - step0: Calculate: \(\frac{69000\times 500^{8}}{523^{8}}\) Brandon would need to invest approximately $48,149.92 to reach a value of $69,000 in 8 years with an annual interest rate of 4.6% compounded annually.