Sure! Let's break down the problem step by step: --- ## **Step 1: Find the Additional Cost of Upgrading to R-19 Insulation** - **Area to be insulated:** 1,450 sq ft - **Extra cost for upgrade:** $.05/sq ft \[ \text{Extra cost} = 145 \times .05 = \$72.50 \] --- ## **Step 2: Find the Annual Savings** - **Annual energy saved:** 370 kWh - **Electricity cost:** $.05/kWh \[ \text{Annual savings} = 370 \times .05 = \$18.50 \] --- ## **Step 3: Find the Future Worth of Savings Over 25 Years** - **Annual savings:** $18.50 - **Number of years (n):** 25 - **Interest rate (i):** 11% per year The future worth (FW) of an **annual series** is: \[ FW = A \cdot \frac{(1+i)^n - 1}{i} \] Where: - \( A \) = Annual amount ($18.50) - \( i \) = Interest rate (.11) - \( n \) = Number of years (25) Plug in the numbers: \[ FW = 18.50 \times \frac{(1+.11)^{25} - 1}{.11} \] Calculate \((1+.11)^{25}\): \[ (1.11)^{25} \approx 12.23698 \] \[ FW = 18.50 \times \frac{12.23698 - 1}{.11} \] \[ FW = 18.50 \times \frac{11.23698}{.11} \] \[ FW = 18.50 \times 102.15436 \] \[ FW = \$1,890.86 \] --- ## **Step 4: Subtract the Future Worth of the Initial Cost** The initial extra cost is $72.50, but we need to bring this to future value in 25 years: \[ \text{Future Value of Cost} = \text{Present Cost} \times (1+i)^n \] \[ = 72.50 \times 12.23698 = \$887.68 \] --- ## **Step 5: Net Future Worth** \[ \text{Net Future Worth} = \text{FW of Savings} - \text{FW of Cost} \] \[ = 1,890.86 - 887.68 = \$1,003.18 \] --- ## **Step 6: Decision** Since the net future worth is **positive**, the upgrade **is** economically justified. --- ### **Summary Table** | Item | Calculation | Value | |---------------------|---------------------------|------------| | Extra cost | 145 × $.05 | $72.50 | | Annual savings | 370 × $.05 | $18.50 | | FW of savings | See formula above | $1,890.86 | | FW of cost | $72.50 × (1.11)^25 | $887.68 | | **Net FW** | $1,890.86 - $887.68 | $1,003.18 | --- ### **Final Answers** - **The future worth of R-19 insulation is:** **\$1,003.18** - **R-19 insulation is attractive.** --- Let me know if you want to see any step in more detail or need help with the calculator!