Khloe is going to invest in an account paying an interest rate of \( 1.5 \% \) compounded annually. How much would Khloe need to invest, to the nearest dollar, for the value of the account to reach \( \$ 54,000 \) in 11 years?

To find out how much Khloe needs to invest to reach a value of $54,000 in 11 years with an annual interest rate of 1.5% 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 (in %), - \( n \) is the number of years. Given: - \( A = \$54,000 \), - \( r = 1.5\% \), - \( n = 11 \) years. We need to solve for \( P \) to find out how much Khloe needs to invest. Solve the equation by following steps: - step0: Solve for \(P\): \(54000=P\left(1+\frac{1.5}{100}\right)^{11}\) - step1: Simplify: \(54000=\frac{203^{11}}{200^{11}}\times P\) - step2: Swap the sides: \(\frac{203^{11}}{200^{11}}\times P=54000\) - step3: Rewrite the expression: \(\frac{203^{11}P}{200^{11}}=54000\) - step4: Cross multiply: \(203^{11}P=200^{11}\times 54000\) - step5: Simplify the equation: \(203^{11}P=54000\times 200^{11}\) - step6: Divide both sides: \(\frac{203^{11}P}{203^{11}}=\frac{54000\times 200^{11}}{203^{11}}\) - step7: Divide the numbers: \(P=\frac{54000\times 200^{11}}{203^{11}}\) The amount Khloe needs to invest to reach a value of $54,000 in 11 years with an annual interest rate of 1.5% compounded annually is: \[ P = \frac{54000 \times 200^{11}}{203^{11}} \] To find the value of \( P \) to the nearest dollar, we can calculate it. Calculate the value by following steps: - step0: Calculate: \(\frac{54000\times 200^{11}}{203^{11}}\) Khloe needs to invest approximately $45,842.39 to reach a value of $54,000 in 11 years with an annual interest rate of 1.5% compounded annually.