Let's break down the problem step-by-step: ### Given Data - **First cost (Initial Investment), \( P \):** \$60,000 (outflow, so negative) - **Useful life:** 15 years - **Salvage value:** \$0 - **Labor costs increase:** \$2,250 per year (outflow) - **Raw material costs decrease:** \$11,500 per year (inflow) - **MARR:** 5%/year (for comparison, not needed for IRR calculation) - **Net annual cash flow:** ? ### Step 1: Calculate Net Annual Cash Flow \[ \text{Net annual cash flow} = \text{Savings} - \text{Extra costs} \] \[ = \$11,500 - \$2,250 = \$9,250 \quad (\text{inflow, positive}) \] ### Step 2: Set up the IRR Equation The cash flow profile is: - Year 0: \(-\$60,000\) - Years 1-15: \(+\$9,250\) per year We want to find the interest rate \( i \) such that: \[ 0 = -60,000 + 9,250 \cdot (P/A, i, 15) \] Where \((P/A, i, 15)\) is the present worth factor for an ordinary annuity. The present worth factor is: \[ (P/A, i, n) = \frac{1 - (1+i)^{-n}}{i} \] So, \[ -60,000 + 9,250 \left[ \frac{1 - (1+i)^{-15}}{i} \right] = 0 \] \[ 9,250 \left[ \frac{1 - (1+i)^{-15}}{i} \right] = 60,000 \] \[ \frac{1 - (1+i)^{-15}}{i} = \frac{60,000}{9,250} = 6.486486 \] ### Step 3: Solve for IRR (i) We need to find \( i \) such that: \[ \frac{1 - (1+i)^{-15}}{i} = 6.486486 \] This requires either trial and error, a financial calculator, or Excel. #### Let's try by using the Present Worth Factor Table or Excel formula: Alternatively, let's estimate by trying different interest rates. #### Try \( i = 8\% \): \[ (P/A, 8\%, 15) = \frac{1 - (1+0.08)^{-15}}{0.08} \] \[ (1+0.08)^{15} = 3.1722 \implies (1+0.08)^{-15} = 1/3.1722 = 0.3152 \] \[ \frac{1 - 0.3152}{0.08} = \frac{0.6848}{0.08} = 8.56 \] This is more than 6.486; so we need a higher \( i \). #### Try \( i = 12\% \): \[ (1+0.12)^{15} = 5.473 \implies (1+0.12)^{-15} = 1 / 5.473 = 0.1827 \] \[ \frac{1 - 0.1827}{0.12} = \frac{0.8173}{0.12} = 6.81 \] Still a bit high. #### Try \( i = 13\% \): \[ (1+0.13)^{15} = (approx) 6.144 \implies (1+0.13)^{-15} = 1/6.144 = 0.1627 \] \[ 1 - 0.1627 = 0.8373 \] \[ 0.8373 / 0.13 = 6.44 \] Now, 6.44 < 6.486. So the IRR is between 12% and 13%. #### Interpolation Let: - At 12%: 6.81 - At 13%: 6.44 We want 6.486. \[ i = 12\% + \frac{6.81 - 6.486}{6.81 - 6.44} \times (13\% - 12\%) \] \[ = 12\% + \frac{0.324}{0.37} \times 1\% \] \[ = 12\% + 0.876 \% \] \[ = 12.88\% \] ### Step 4: **Final Answer** **The internal rate of return (IRR) of this investment is approximately:** \[ \boxed{12.9\%} \] --- **Summary of steps:** 1. Calculate net annual cash flow = \$9,250 2. Set up the IRR equation with present worth factor. 3. Plug in values, solve for the interest rate by trial and interpolation. 4. Final answer: **IRR ≈ 12.9%**