Present value (with changing interest rates). Marty has been offered an injury settlement of \( \$ 10,000 \) payable in 3 years. He wants to know what the present value of the injury settlement is if his opportunit) cost is \( 4 \% \). (The opportunity cost is the interest rate in this problem.) What if the opportunity cost is \( 6.5 \% \) ? What if it is \( 10.5 \% \) ? If Marty's opportunity cost is \( 6.5 \% \), what is the present value of the injury settlement? \( \$ 8278.49 \) (Round to the nearest cent.) If Marty's opportunity cost is \( 10.5 \% \), what is the present value of the injury settlement? \( \$ \square \) (Round to the nearest cent.)

To find the present value of the injury settlement, we can use the formula for present value: \[ PV = \frac{FV}{(1 + r)^n} \] where: - \( PV \) is the present value, - \( FV \) is the future value (in this case, the injury settlement of $10,000), - \( r \) is the opportunity cost (interest rate), - \( n \) is the number of years until the settlement is paid. Given that Marty's opportunity cost is 4%, 6.5%, and 10.5%, we can calculate the present value for each interest rate. 1. For an opportunity cost of 4%: \[ PV = \frac{10000}{(1 + 0.04)^3} \] 2. For an opportunity cost of 6.5%: \[ PV = \frac{10000}{(1 + 0.065)^3} \] 3. For an opportunity cost of 10.5%: \[ PV = \frac{10000}{(1 + 0.105)^3} \] Let's calculate the present value for each opportunity cost. Calculate the value by following steps: - step0: Calculate: \(\frac{10000}{\left(1+0.04\right)^{3}}\) - step1: Add the numbers: \(\frac{10000}{1.04^{3}}\) - step2: Convert the expressions: \(\frac{10000}{\left(\frac{26}{25}\right)^{3}}\) - step3: Simplify the expression: \(\frac{10000}{\frac{26^{3}}{25^{3}}}\) - step4: Rewrite the expression: \(\frac{156250000}{26^{3}}\) - step5: Reduce the fraction: \(\frac{19531250}{2197}\) Calculate or simplify the expression \( 10000/(1+0.065)^3 \). Calculate the value by following steps: - step0: Calculate: \(\frac{10000}{\left(1+0.065\right)^{3}}\) - step1: Add the numbers: \(\frac{10000}{1.065^{3}}\) - step2: Convert the expressions: \(\frac{10000}{\left(\frac{213}{200}\right)^{3}}\) - step3: Simplify the expression: \(\frac{10000}{\frac{213^{3}}{200^{3}}}\) - step4: Rewrite the expression: \(\frac{10000\times 200^{3}}{213^{3}}\) Calculate or simplify the expression \( 10000/(1+0.105)^3 \). Calculate the value by following steps: - step0: Calculate: \(\frac{10000}{\left(1+0.105\right)^{3}}\) - step1: Add the numbers: \(\frac{10000}{1.105^{3}}\) - step2: Convert the expressions: \(\frac{10000}{\left(\frac{221}{200}\right)^{3}}\) - step3: Simplify the expression: \(\frac{10000}{\frac{221^{3}}{200^{3}}}\) - step4: Rewrite the expression: \(\frac{10000\times 200^{3}}{221^{3}}\) The present value of the injury settlement for an opportunity cost of 4% is approximately $8,889.96. The present value of the injury settlement for an opportunity cost of 6.5% is approximately $8,278.49. The present value of the injury settlement for an opportunity cost of 10.5% is approximately $7,411.62.