Victoria Morefield 02/03/25 5:02 AM ce-compound Cuastion 18, \( 4,8.84 \) HWS Scces \( 40 \% 8 \) of 20 polnts Politss0 of 1 Save Suppose you start saving today for a \( \$ 30,000 \) down payment that you plan to mako on a house in 7 years. Assurre hal you make no deposits into the account after the initial deposit. For the accounl described below, how much would you have to deposit now to reach your \( \$ 30,000 \) goal in 7 years. An account wilt daily compounding and an APR of \( 8 \% \) You should invest \( \$ \) \( \square \) (Do not round until the final answer. Nenen raurd to two docimal paces as reeted.)

To determine how much you need to deposit now to reach your $30,000 goal in 7 years with daily compounding and an APR of 8%, we can use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( A \) is the future value of the investment (in this case, $30,000) - \( P \) is the principal amount (the initial deposit) - \( r \) is the annual interest rate (8% in this case) - \( n \) is the number of times the interest is compounded per year (365 for daily compounding) - \( t \) is the time in years (7 years in this case) We need to solve for \( P \) to find out how much you need to deposit now to reach your goal in 7 years. Calculate the value by following steps: - step0: Calculate: \(30000\left(1+\frac{8}{365}\right)^{365}\times 7\) - step1: Add the numbers: \(30000\left(\frac{373}{365}\right)^{365}\times 7\) - step2: Multiply the terms: \(210000\left(\frac{373}{365}\right)^{365}\) - step3: Simplify: \(210000\times \frac{373^{365}}{365^{365}}\) - step4: Rewrite the expression: \(625\times 336\times \frac{373^{365}}{365^{365}}\) - step5: Rewrite the expression: \(625\times 336\times \frac{373^{365}}{5^{365}\times 73^{365}}\) - step6: Rewrite the expression: \(5^{4}\times 336\times \frac{373^{365}}{5^{365}\times 73^{365}}\) - step7: Reduce the numbers: \(336\times \frac{373^{365}}{5^{361}\times 73^{365}}\) - step8: Multiply the fractions: \(\frac{336\times 373^{365}}{5^{361}\times 73^{365}}\) The amount you need to deposit now to reach your $30,000 goal in 7 years with daily compounding and an APR of 8% is approximately $5,741,791. Therefore, you would need to deposit $5,741,791 now to reach your $30,000 goal in 7 years.