6 Formula 1 point You will need to make a payment of \( \$ 4,000 \) ecactly 6 years from today. How much should you deposit today in a bank account that pays an interest rate of \( 7.2 \% \) with annual compounding to ensure you will have enough money to make the future payment? Round your answer to the nearest penny. Type your answer-

To find out how much you should deposit today in a bank account that pays an interest rate of 7.2% with annual compounding to ensure you will have enough money to make the future payment of $4,000 exactly 6 years from today, we can use the formula for compound interest: \[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \] Where: - \( A \) is the future amount (in this case, $4,000) - \( P \) is the principal amount (the amount you need to deposit today) - \( r \) is the annual interest rate (7.2% in this case) - \( n \) is the number of times the interest is compounded per year (1 for annual compounding) - \( t \) is the time in years (6 years in this case) We can rearrange the formula to solve for \( P \) (the principal amount): \[ P = \frac{A}{\left( 1 + \frac{r}{n} \right)^{nt}} \] Substitute the given values into the formula: \[ P = \frac{4000}{\left( 1 + \frac{0.072}{1} \right)^{1 \times 6}} \] Now, we can calculate the value of \( P \) to determine how much you should deposit today. Calculate the value by following steps: - step0: Calculate: \(\frac{4000}{\left(1+0.072\right)^{6}}\) - step1: Add the numbers: \(\frac{4000}{1.072^{6}}\) - step2: Convert the expressions: \(\frac{4000}{\left(\frac{134}{125}\right)^{6}}\) - step3: Simplify the expression: \(\frac{4000}{\frac{134^{6}}{125^{6}}}\) - step4: Rewrite the expression: \(\frac{4000\times 125^{6}}{134^{6}}\) - step5: Reduce the fraction: \(\frac{125^{7}}{180916764338}\) The amount you should deposit today in a bank account that pays an interest rate of 7.2% with annual compounding to ensure you will have enough money to make the future payment of $4,000 exactly 6 years from today is approximately $2,635.67. Therefore, you should deposit $2,635.67 today in the bank account to ensure you will have enough money to make the future payment of $4,000 exactly 6 years from today.