Business applications: cost, revenue and profit
Problem 3.655 · medium
A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{200} + 5 x + 500 \) dollars, and each unit sells for \( \displaystyle \$25 \). Find the marginal cost at \( \displaystyle x = 10 \), and the number of units that maximizes profit, and that profit.
- \[ \left. \frac{d}{d x} \left(\frac{x^{2}}{200} + 5 x + 500\right) \right|_{\substack{ x=10 }} = \frac{51}{10} \]Marginal cost C′(10): about the cost of unit 11.✓ Proved
- \[ 25 x - \left(\frac{x^{2}}{200} + 5 x + 500\right) = - \frac{x^{2}}{200} + 20 x - 500 \]Profit = revenue − cost.✓ Proved
- \[ \frac{d}{d x} \left(- \frac{x^{2}}{200} + 20 x - 500\right) = 20 - \frac{x}{100} \]P′(x).✓ Proved
- \[ \left. 20 - \frac{x}{100} \right|_{\substack{ x=2000 }} = 0 \]P′ = 0 at x = 2000.✓ Proved
- \[ \frac{d^{2}}{d x^{2}} \left(- \frac{x^{2}}{200} + 20 x - 500\right) = - \frac{1}{100} \]P″ < 0: a maximum.✓ Proved
- \[ \left. 20 x + \frac{\left(-1\right) x^{2}}{200} - 500 \right|_{\substack{ x=2000 }} = 19500 \]The maximum profit.✓ Proved
Answer \( C'(10) = \frac{51}{10},\ x = 2000,\ P(2000) = 19500 \)
Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | profit tried at every whole number of units from 0 to 20000; marginal cost by a central difference |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the marginal cost, sets up the profit function, finds the critical point, verifies it is a maximum using the second derivative test, and calculates the maximum profit. The interpretation of marginal cost in line 1 is standard and acceptable.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the marginal cost, sets up the profit function, finds the critical point, verifies it is a maximum using the second derivative test, and calculates the maximum profit. The interpretation of marginal cost in line 1 is standard and acceptable.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: fail (misleading) 2026-10-14 — The comment in line 1 claims marginal cost is 'about the cost of unit 11', which is imprecise and potentially misleading; marginal cost C'(x) is the instantaneous rate of change, approximating the cost of the (x+1)-th unit, but for x=10 it approximates the cost of the 11th unit only if interpreted loosely, yet standard interpretation is the cost to produce one additional unit at that rate. More critically, the phrasing 'about the cost of unit 11' suggests it equals the actual cost of that specific unit, which is false (C(11)-C(10) != C'(10)). This teaches a false distinction between marginal cost and discrete unit cost.gpt-oss:20b: pass 2026-10-14
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/marginal_analysis, checked 2026-10-17 with SymPy 1.14.0.