If money can be invested at 6 . \( 3 \% \) compounded quarterly, which is larger, \( \$ 8619 \) now or \( \$ 16,000 \) in 10 years? Use present value to decide. The present value of \( \$ 16,000 \) in 10 years is \( \$ \). (Do not round until the final answer. Then round to the nearest cent as needed)

To determine which amount is larger, we need to calculate the present value of $16,000 in 10 years using the formula for present value: \[ PV = \frac{FV}{(1 + r)^n} \] where: - \( PV \) is the present value, - \( FV \) is the future value ($16,000 in this case), - \( r \) is the annual interest rate (6.3% in this case), - \( n \) is the number of years (10 years in this case). Given that the interest is compounded quarterly, we need to adjust the annual interest rate to a quarterly rate. The formula for quarterly interest rate is: \[ r_{quarterly} = \frac{r_{annual}}{4} \] Substitute the values into the formula for present value: \[ PV = \frac{16,000}{(1 + \frac{6.3}{4})^{10}} \] Now, we can calculate the present value of $16,000 in 10 years. Calculate the value by following steps: - step0: Calculate: \(\frac{16000}{\left(1+\frac{6.3}{4}\right)^{10}}\) - step1: Divide the terms: \(\frac{16000}{\left(1+\frac{63}{40}\right)^{10}}\) - step2: Add the numbers: \(\frac{16000}{\left(\frac{103}{40}\right)^{10}}\) - step3: Simplify the expression: \(\frac{16000}{\frac{103^{10}}{40^{10}}}\) - step4: Rewrite the expression: \(\frac{16000\times 40^{10}}{103^{10}}\) The present value of $16,000 in 10 years is approximately $1,248.382. Now, we need to compare this present value with $8,619 to determine which amount is larger. Numbers by following steps: - step0: Solution: \(8619\) The present value of $8,619 is $8,619. Comparing the present values, $8,619 is larger than $1,248.382. Therefore, $8,619 is larger than $16,000 in 10 years.