Business applications: cost, revenue and profit
Problem 3.681 · medium
A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{200} + 5 x + 100 \) dollars, and each unit sells for \( \displaystyle \$65 \). Find the marginal cost at \( \displaystyle x = 10 \), and the number of units that maximizes profit, and that profit.
- \[ \left. \frac{d}{d x} \left(\frac{x^{2}}{200} + 5 x + 100\right) \right|_{\substack{ x=10 }} = \frac{51}{10} \]Marginal cost C′(10): about the cost of unit 11.✓ Proved
- \[ 65 x - \left(\frac{x^{2}}{200} + 5 x + 100\right) = - \frac{x^{2}}{200} + 60 x - 100 \]Profit = revenue − cost.✓ Proved
- \[ \frac{d}{d x} \left(- \frac{x^{2}}{200} + 60 x - 100\right) = 60 - \frac{x}{100} \]P′(x).✓ Proved
- \[ \left. 60 - \frac{x}{100} \right|_{\substack{ x=6000 }} = 0 \]P′ = 0 at x = 6000.✓ Proved
- \[ \frac{d^{2}}{d x^{2}} \left(- \frac{x^{2}}{200} + 60 x - 100\right) = - \frac{1}{100} \]P″ < 0: a maximum.✓ Proved
- \[ \left. 60 x + \frac{\left(-1\right) x^{2}}{200} - 100 \right|_{\substack{ x=6000 }} = 179900 \]The maximum profit.✓ Proved
Answer \( C'(10) = \frac{51}{10},\ x = 6000,\ P(6000) = 179900 \)
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: fail (misleading) — The comment in line 1 incorrectly defines marginal cost as 'about the cost of unit 11'. Marginal cost at x=10 is the approximate cost of producing the 11th unit, but strictly speaking, it is the instantaneous rate of change at x=10, not the cost of a specific discrete unit. While often used as an approximation, stating it as 'the cost of unit 11' is imprecise and potentially misleading in a calculus context where the distinction between discrete and continuous models matters.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-18 — The comment in line 1 incorrectly defines marginal cost as 'about the cost of unit 11'. Marginal cost at x=10 is the approximate cost of producing the 11th unit, but strictly speaking, it is the instantaneous rate of change at x=10, not the cost of a specific discrete unit. While often used as an approximation, stating it as 'the cost of unit 11' is imprecise and potentially misleading in a calculus context where the distinction between discrete and continuous models matters.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly derives the profit function, identifies the critical point, verifies it is a maximum using the second derivative test, and calculates the correct values for marginal cost and maximum profit.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-14gpt-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 2026-10-18 with SymPy 1.14.0.