Let's break down the problem step by step: ## **Given Data** - **Cars per restaurant:** 2 - **Car cost:** \$23,000 each - **Total car cost:** 2 × \$23,000 = **\$46,000** - **Depreciation:** Straight line over **3 years** to a salvage value of **\$5,000 (at the end of year 5, not depreciated)** - **Annual operating costs:** **\$18,000** - **Variable cost per delivery:** **\$1.50** - **Average charge per delivery:** **\$3.00** - **Tax rate:** **21%** - **Discount rate:** **8%** - **Salvage value realized in year 5, non-taxable (\$5,000)** Let **Q** be the number of meals delivered **per year**. --- ## **1. Depreciation Calculation** - **Depreciable amount:** \$46,000 (cost) – \$5,000 (salvage) = **\$41,000** - **Annual depreciation:** \$41,000 / 3 = **\$13,666.67/year** (for years 1-3 only) --- ## **2. Annual Profit Equation (Accounting Approach)** **Annual Revenue:** \[ \text{Revenue} = Q \times \$3.00 \] **Annual Costs (excluding depreciation):** - Operating cost: \$18,000 - Variable cost: \$1.50 × Q - Depreciation: \$13,666.67 (years 1-3 only) **EBIT (Earnings Before Interest and Tax):** \[ \text{EBIT} = (\$3.00 - \$1.50)Q - \$18,000 - \$13,666.67 \] \[ \text{EBIT} = \$1.50Q - \$31,666.67 \] **Tax:** \[ \text{Tax} = 21\% \times \text{EBIT} \] (Break-even means EBIT = 0, so tax = 0) **Net Income (at accounting break-even):** Set EBIT = 0: \[ \$1.50Q - \$31,666.67 = 0 \] \[ Q = \frac{31,666.67}{1.50} = 21,111 \] **Meals per day:** \[ \frac{21,111}{365} \approx \boxed{58} \] --- ## **3. Cash Flow Approach (Present Value or NPV Break-Even)** We need to find the value of **Q** that yields NPV = 0 over the life of the project. Let's **assume a project life of 5 years** (since salvage is realized in year 5). ### **Cash Flows:** - **Initial Outlay (Year 0):** **-\$46,000** - **Annual operating cash flow (Years 1–5):** - Revenue: \$3.00Q - Operating cost: \$18,000 - Variable cost: \$1.50Q - Depreciation: \$13,666.67 (years 1–3), \$0 (years 4–5) - Tax: 21% - **Salvage Value (Year 5):** **\$5,000** (not taxed) #### **Calculate After-tax Cash Flows:** **Years 1–3:** - EBIT = \$1.50Q - \$31,666.67 - Tax = 0.21 × EBIT - Net income = EBIT × (1 - 0.21) - Add back depreciation (non-cash): \[ \text{After-tax cash flow} = (\$1.50Q - \$31,666.67) \times (1 - 0.21) + \$13,666.67 \] \[ = (\$1.50Q - \$31,666.67) \times 0.79 + \$13,666.67 \] \[ = 1.185Q - 25,016.67 + 13,666.67 \] \[ = 1.185Q - 11,350 \] **Years 4–5:** - No depreciation. - EBIT = \$1.50Q - \$18,000 - After-tax EBIT = (\$1.50Q - \$18,000) × 0.79 \[ = 1.185Q - 14,220 \] **Salvage in Year 5:** +\$5,000 --- ### **Set Up the NPV Equation** \[ 0 = -46,000 + \sum_{t=1}^{3} \frac{(1.185Q - 11,350)}{(1.08)^t} + \sum_{t=4}^{5} \frac{(1.185Q - 14,220)}{(1.08)^t} + \frac{5,000}{(1.08)^5} \] Let's calculate the discount factors: - (1.08)^1 = 1.08 - (1.08)^2 = 1.1664 - (1.08)^3 = 1.2597 - (1.08)^4 = 1.3605 - (1.08)^5 = 1.4693 \[ \frac{1}{1.08^1} = 0.9259 \] \[ \frac{1}{1.08^2} = 0.8573 \] \[ \frac{1}{1.08^3} = 0.7938 \] \[ \frac{1}{1.08^4} = 0.7350 \] \[ \frac{1}{1.08^5} = 0.6806 \] Now, let's calculate the present value of each year's cash flow **per 1 unit**, then sum coefficients for Q and constants. #### **For Q Terms:** **Years 1–3:** \[ 1.185Q \times (0.9259 + 0.8573 + 0.7938) = 1.185Q \times 2.577 \] \[ = 3.052Q \] **Years 4–5:** \[ 1.185Q \times (0.7350 + 0.6806) = 1.185Q \times 1.4156 \] \[ = 1.678Q \] **Total Q coefficient:** 3.052Q + 1.678Q = **4.73Q** --- #### **For Constant Terms:** **Years 1–3:** \[ -11,350 \times (0.9259 + 0.8573 + 0.7938) = -11,350 \times 2.577 = -29,247 \] **Years 4–5:** \[ -14,220 \times (0.7350 + 0.6806) = -14,220 \times 1.4156 = -20,163 \] **Salvage Value (Year 5):** \[ \frac{5,000}{1.4693} = 3,403 \] **Total constants:** \[ -29,247 - 20,163 + 3,403 = -46,007 \] --- #### **Now the NPV Equation:** \[ 0 = -46,000 + 4.73Q - 46,007 \] \[ 0 = 4.73Q - 92,007 \] \[ 4.73Q = 92,007 \] \[ Q = \frac{92,007}{4.73} = 19,452 \] --- ### **Number of Deliveries per Day** \[ \frac{19,452}{365} = \boxed{53} \] --- ## **Summary Table** | Approach | Deliveries per Year | Deliveries per Day | |--------------------------|--------------------|-------------------| | **Accounting Break-even**| 21,111 | 58 | | **NPV Break-even** | 19,452 | 53 | --- ### **Conclusion:** - **Accounting break-even:** ~58 deliveries per day - **Present value (NPV) break-even:** ~53 deliveries per day **The NPV break-even is lower because it accounts for the time value of money and the salvage value.** --- ### **If you need the math in more detail, or in Excel format, let me know!**