1. **Stating the problem:** We are given data about the quantity of diamonds produced, variable costs, prices, and total revenue. We need to find the marginal cost and marginal revenue for each quantity, determine the profit-maximizing quantity, analyze profit or loss, and make a recommendation. 2. **Formulas and rules:** - Marginal Cost (MC) is the change in total cost when one more unit is produced. Since fixed costs are constant, MC = change in variable cost between quantities. - Marginal Revenue (MR) is the change in total revenue when one more unit is sold. - Profit maximization occurs where MR = MC or where MR just exceeds MC before MR < MC. - Profit or loss depends on total revenue minus total cost (fixed + variable). 3. **Calculate Marginal Cost (MC):** Given variable costs (VC), MC for quantity $q$ is: $$MC_q = VC_q - VC_{q-1}$$ | Quantity | VC | MC Calculation | MC | |----------|-----|------------------------|------| | 0 | 0 | - | - | | 1 | 900 | 900 - 0 | 900 | | 2 | 1750| 1750 - 900 | 850 | | 3 | 2550| 2550 - 1750 | 800 | | 4 | 3400| 3400 - 2550 | 850 | | 5 | 4310| 4310 - 3400 | 910 | | 6 | 5260| 5260 - 4310 | 950 | | 7 | 6250| 6250 - 5260 | 990 | | 8 | 7310| 7310 - 6250 | 1060 | | 9 | 8460| 8460 - 7310 | 1150 | | 10 | 9720| 9720 - 8460 | 1260 | 4. **Calculate Marginal Revenue (MR):** Given total revenue (TR), MR for quantity $q$ is: $$MR_q = TR_q - TR_{q-1}$$ | Quantity | TR | MR Calculation | MR | |----------|------|-----------------------|------| | 0 | 0 | - | - | | 1 | 1450 | 1450 - 0 | 1450 | | 2 | 2800 | 2800 - 1450 | 1350 | | 3 | 4050 | 4050 - 2800 | 1250 | | 4 | 5200 | 5200 - 4050 | 1150 | | 5 | 6250 | 6250 - 5200 | 1050 | | 6 | 7200 | 7200 - 6250 | 950 | | 7 | 8050 | 8050 - 7200 | 850 | | 8 | 8800 | 8800 - 8050 | 750 | | 9 | 9450 | 9450 - 8800 | 650 | | 10 | 10000| 10000 - 9450 | 550 | 5. **Determine profit-maximizing quantity:** Profit maximization occurs where MR $\\geq$ MC and MR next quantity < MC next quantity. Compare MC and MR: - At quantity 5: MC = 910, MR = 1050 (MR > MC) - At quantity 6: MC = 950, MR = 950 (MR = MC) - At quantity 7: MC = 990, MR = 850 (MR < MC) So, the profit-maximizing quantity is **6 diamonds**. 6. **Profit or loss at quantity 6:** - Total cost = Fixed cost + Variable cost = $500 + 5260 = 5760$ - Total revenue = 7200 - Profit = Total revenue - Total cost = 7200 - 5760 = 1440 Since profit is positive, the mine is experiencing a **profit** at quantity 6. 7. **Recommendation:** Given the positive profit at quantity 6, the mine should **continue production at this level** to maximize profit. Final answers: - Marginal Cost and Marginal Revenue as calculated above. - Profit-maximizing quantity: 6 - The mine is experiencing a profit. - Recommendation: Continue production at quantity 6.