An account pays \( 9.4 \% \) per year Suppose 18 years from now you would like to have \( \$ 7000 \) in the account, with no additional money added to the account other than interest. (a) If it pays \( 9.4 \% \) compounded annually, what minimum amount must be deposited into this account today in order to have \( \$ 7000 \) in the account 18 years from now? \( \$ \) Number (Round to the nearest 0.01 dollars) (b) If it pays \( 9.4 \% \) compounded weekly ( 52 times per year), what minimum amount must be deposited into this account today in order to have \( \$ 7000 \) in the account 18 years from now? \( \$ \) Number (Round to the nearest 0.01 dollars) (c) If it pays \( 9.4 \% \) compounded continuously, what minimum amount must be deposited into this account today in order to have \( \$ 7000 \) in the account 18 years from now? \( \$ \) Number (Round to the nearest 0.01 dollars)

To determine the minimum amount you need to deposit today, we can use the formula for future value based on compound interest. We can break down the calculations for each compounding method. (a) For annual compounding, the formula is: \[ PV = \frac{FV}{(1 + r)^t} \] Where \( PV \) is the present value, \( FV \) is the future value (\$7000), \( r \) is the annual interest rate (0.094), and \( t \) is the number of years (18). Calculating, we get: \[ PV = \frac{7000}{(1 + 0.094)^{18}} \approx \frac{7000}{4.487681} \approx 1553.73 \] So, you must deposit approximately **\$1553.73** today. (b) For weekly compounding, we modify the formula to account for the number of compounding periods per year: \[ PV = \frac{FV}{(1 + \frac{r}{n})^{nt}} \] Where \( n \) is the number of compounding periods per year (52). Plugging in the values: \[ PV = \frac{7000}{(1 + \frac{0.094}{52})^{52 \times 18}} \approx \frac{7000}{5.02775} \approx 1397.46 \] Thus, you would need to deposit approximately **\$1397.46** today. (c) For continuous compounding, we use the formula: \[ PV = FV \cdot e^{-rt} \] Where \( e \) is approximately 2.71828. Thus: \[ PV = 7000 \cdot e^{-0.094 \times 18} \approx 7000 \cdot e^{-1.692} \approx 7000 \cdot 0.18466 \approx 1292.25 \] You must deposit approximately **\$1292.25** today for continuous compounding. So, the rounded answers are: (a) \$1553.73 (b) \$1397.46 (c) \$1292.25