## Introduction This problem involves the required annual revenue from a mine investment, so the earns a .5% annual return while also recovering the original investment using a sinking fund that earns 4% per year. The investor pays $395,000, and the mine produces revenue for 24 years, with payments received at the end of each year. --- ## Step-by-Step Solution ### **Step 1: Understand the Cash Flows** - **Initial Investment, \( P \):** $395,000 (paid today) - **Annual Revenue, \( R \):** unknown (received at end of each year for 24 years) - **Required Annual Return (interest rate), \( i \):** 5.5% - **Sinking Fund Rate, \( j \):** 4% - **Number of Years, \( n \):** 24 ### **Step 2: Set Up the Sinking Fund Factor** The annual payment must: - Provide the investor a 5.5% return on the investment. - Accumulate enough in a sinking fund to recover the principal ($395,000) in 24 years at 4%. Let: - \( A_1 \) = annual revenue to provide the 5.5% return (interest portion) - \( A_2 \) = annual deposit into the sinking fund to recover the principal So, **Total Annual Revenue \( R \) = \( A_1 + A_2 \)** --- ### **Step 3: Calculate the Sinking Fund Deposit (\( A_2 \))** The sinking fund deposit should grow to $395,000 in 24 years at 4%. \[ A_2 = \frac{P}{\left[ \frac{(1 + j)^n - 1}{j} \right]} \] \[ A_2 = \frac{395,000}{\left[ \frac{(1 + .04)^{24} - 1}{.04} \right]} \] Calculate denominator: \[ (1 + .04)^{24} = 2.563 \] \[ 2.563 - 1 = 1.563 \] \[ \frac{1.563}{.04} = 39.075 \] So, \[ A_2 = \frac{395,000}{39.075} \approx \$10,112.98 \] --- ### **Step 4: Calculate the Interest Portion (\( A_1 \))** This is simply the required return on the investment each year at 5.5%: \[ A_1 = P \times i = 395,000 \times .055 = \$21,725 \] --- ### **Step 5: Calculate the Total Required Annual Revenue (\( R \))** \[ R = A_1 + A_2 = 21,725 + 10,112.98 = \$31,837.98 \] --- ## **Summary** The level annual revenue the investor should receive is **\$31,837.98** per year. This amount ensures a 5.5% return on the investment and recovers the principal via a sinking fund earning 4% annually over 24 years.