Business applications: cost, revenue and profit
Problem 3.662 · medium
A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{50} + 5 x + 100 \) dollars, and each unit sells for \( \displaystyle \$45 \). Find the marginal cost at \( \displaystyle x = 10 \), and the number of units that maximizes profit, and that profit.
- \[ \left. \frac{d}{d x} \left(\frac{x^{2}}{50} + 5 x + 100\right) \right|_{\substack{ x=10 }} = \frac{27}{5} \]Marginal cost C′(10): about the cost of unit 11.✓ Proved
- \[ 45 x - \left(\frac{x^{2}}{50} + 5 x + 100\right) = - \frac{x^{2}}{50} + 40 x - 100 \]Profit = revenue − cost.✓ Proved
- \[ \frac{d}{d x} \left(- \frac{x^{2}}{50} + 40 x - 100\right) = 40 - \frac{x}{25} \]P′(x).✓ Proved
- \[ \left. 40 - \frac{x}{25} \right|_{\substack{ x=1000 }} = 0 \]P′ = 0 at x = 1000.✓ Proved
- \[ \frac{d^{2}}{d x^{2}} \left(- \frac{x^{2}}{50} + 40 x - 100\right) = - \frac{1}{25} \]P″ < 0: a maximum.✓ Proved
- \[ \left. 40 x + \frac{\left(-1\right) x^{2}}{50} - 100 \right|_{\substack{ x=1000 }} = 19900 \]The maximum profit.✓ Proved
Answer \( C'(10) = \frac{27}{5},\ x = 1000,\ P(1000) = 19900 \)
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 (style) — [domain objection, downgraded to style] The comment in line 1 claims marginal cost is 'about the cost of unit 11', which is incorrect; C'(10) approximates the cost of the 11th unit (going from 10 to 11), but the phrasing is imprecise and potentially confusing regarding discrete vs continuous interpretation. More importantly, the problem asks for 'the number of units that maximizes profit', and while x=1000 is the critical point, the solution does not explicitly verify that this is a global maximum on the domain x>=0, though the second derivative test confirms it is a local maximum for a downward parabola, which is sufficient. However, the primary issue is the misleading comment in line 1.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-17 — [domain objection, downgraded to style] The comment in line 1 claims marginal cost is 'about the cost of unit 11', which is incorrect; C'(10) approximates the cost of the 11th unit (going from 10 to 11), but the phrasing is imprecise and potentially confusing regarding discrete vs continuous interpretation. More importantly, the problem asks for 'the number of units that maximizes profit', and while x=1000 is the critical point, the solution does not explicitly verify that this is a global maximum on the domain x>=0, though the second derivative test confirms it is a local maximum for a downward parabola, which is sufficient. However, the primary issue is the misleading comment in line 1.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (misleading) 2026-10-14 — The comment in line 1 claiming marginal cost is 'about the cost of unit 11' is incorrect; C'(10) approximates the cost of the 11th unit (going from 10 to 11), not the cost of unit 11 itself (which would be C(11)).gpt-oss:20b: pass 2026-10-14qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly calculates marginal cost, derives the profit function, finds the critical point, verifies it is a maximum using the second derivative test, and computes the maximum profit. The interpretation of marginal cost is standard.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.