Business applications: cost, revenue and profit
Problem 3.732 · medium
A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{200} + 10 x + 1000 \) dollars, and each unit sells for \( \displaystyle \$40 \). Find the marginal cost at \( \displaystyle x = 20 \), and the number of units that maximizes profit, and that profit.
- \[ \left. \frac{d}{d x} \left(\frac{x^{2}}{200} + 10 x + 1000\right) \right|_{\substack{ x=20 }} = \frac{51}{5} \]Marginal cost C′(20): about the cost of unit 21.✓ Proved
- \[ 40 x - \left(\frac{x^{2}}{200} + 10 x + 1000\right) = - \frac{x^{2}}{200} + 30 x - 1000 \]Profit = revenue − cost.✓ Proved
- \[ \frac{d}{d x} \left(- \frac{x^{2}}{200} + 30 x - 1000\right) = 30 - \frac{x}{100} \]P′(x).✓ Proved
- \[ \left. 30 - \frac{x}{100} \right|_{\substack{ x=3000 }} = 0 \]P′ = 0 at x = 3000.✓ Proved
- \[ \frac{d^{2}}{d x^{2}} \left(- \frac{x^{2}}{200} + 30 x - 1000\right) = - \frac{1}{100} \]P″ < 0: a maximum.✓ Proved
- \[ \left. 30 x + \frac{\left(-1\right) x^{2}}{200} - 1000 \right|_{\substack{ x=3000 }} = 44000 \]The maximum profit.✓ Proved
Answer \( C'(20) = \frac{51}{5},\ x = 3000,\ P(3000) = 44000 \)
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 claiming the marginal cost is 'about the cost of unit 21' is incorrect; C'(20) approximates the cost of the 21st unit, not the cost of unit 21 itself (which would be C(21)).
Every verdict on record (6)
qwen3.6:27b-mlx: fail (misleading) 2026-10-19 — The comment in line 1 claiming the marginal cost is 'about the cost of unit 21' is incorrect; C'(20) approximates the cost of the 21st unit, not the cost of unit 21 itself (which would be C(21)).gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (style) 2026-10-18 — [domain objection, downgraded to style] The comment on line 1 claims the marginal cost is 'about the cost of unit 21'. This is incorrect; marginal cost at x=20 approximates the cost of the 21st unit (going from 20 to 21), but standard interpretation is the cost of producing one additional unit at that rate, often associated with the 21st unit in discrete terms, but phrasing it as 'cost of unit 21' is ambiguous and potentially misleading regarding the definition of marginal cost (which is instantaneous rate of change). More importantly, the problem asks for 'the number of units that maximizes profit'. While x=3000 is the critical point, the solution does not explicitly verify that this is a global maximum within the feasible domain (x >= 0), though the second derivative test confirms it is a local maximum and the parabola opens downward, implying it is the global maximum. However, the primary issue is the potentially confusing comment in line 1. Let's look closer. Actually, the comment 'about the cost of unit 21' is a common heuristic explanation for C'(20) ~ C(21)-C(20). It is not strictly 'wrong' in a heuristic sense, but is it 'misleading'? A student might think C'(20) IS the cost of the 21st unit. It's an approximation. Is this severe enough to fail? The prompt asks to find what is wrong. The math is correct. The comment is a bit loose. Let's check the rest. Line 5 checks P'' < 0. This confirms a local max. Since P(x) is a downward parabola, this is the global max. The solution is mathematically sound. The only potential issue is the comment in line 1. Is it 'misleading'? It says 'about the cost of unit 21'. This is a standard interpretation in introductory calculus/economics contexts (marginal cost at x approximates cost of x+1-th unit). I will mark it as pass because the mathematical derivation is correct and the comment is a standard, albeit simplified, interpretation.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-18gpt-oss:20b: pass 2026-10-18
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 2026-10-19 with SymPy 1.14.0.