If money can be invested at \( 5.8 \% \) compounded quarterly, which is larger, \( \$ 3855 \) now or \( \$ 6000 \) in 8 years? Use present value to decide. The present value of \( \$ 6000 \) in 8 years is \( \$ \) round to the nearest cent as needed.) (Do not round until the final answer. Then round

To determine which amount is larger, we need to calculate the present value (PV) of \( \$6000 \) that will be received in 8 years, given an interest rate of \( 5.8\% \) compounded quarterly. ### Step 1: Identify the known values - Future Value (FV) = \( 6000 \) - Interest Rate (r) = \( 5.8\% = 0.058 \) - Number of compounding periods per year (n) = \( 4 \) (quarterly) - Time in years (t) = \( 8 \) ### Step 2: Use the present value formula The present value formula for compound interest is given by: \[ PV = \frac{FV}{(1 + \frac{r}{n})^{nt}} \] ### Step 3: Substitute the known values into the formula Now we will substitute the values into the formula: \[ PV = \frac{6000}{(1 + \frac{0.058}{4})^{4 \times 8}} \] ### Step 4: Calculate the present value Let's perform the calculations step by step. 1. Calculate \( \frac{r}{n} \): \[ \frac{0.058}{4} = 0.0145 \] 2. Calculate \( nt \): \[ 4 \times 8 = 32 \] 3. Calculate \( (1 + \frac{r}{n})^{nt} \): \[ (1 + 0.0145)^{32} \] 4. Finally, calculate the present value \( PV \). Now, I will perform these calculations. Calculate the value by following steps: - step0: Calculate: \(\frac{6000}{\left(1+0.0145\right)^{32}}\) - step1: Add the numbers: \(\frac{6000}{1.0145^{32}}\) - step2: Convert the expressions: \(\frac{6000}{\left(\frac{2029}{2000}\right)^{32}}\) - step3: Simplify the expression: \(\frac{6000}{\frac{2029^{32}}{2000^{32}}}\) - step4: Rewrite the expression: \(\frac{6000\times 2000^{32}}{2029^{32}}\) The present value (PV) of \( \$6000 \) in 8 years, calculated using the formula, is approximately \( 3785.17 \). ### Step 5: Compare the present values Now we compare the present value of \( \$6000 \) with \( \$3855 \): - Present Value of \( \$6000 \) in 8 years: \( \$3785.17 \) - Current amount: \( \$3855 \) ### Conclusion Since \( 3855 > 3785.17 \), \( \$3855 \) now is larger than \( \$6000 \) in 8 years. Thus, the present value of \( \$6000 \) in 8 years is \( \$3785.17 \) (rounded to the nearest cent).