Business applications: cost, revenue and profit
Problem 3.659 · medium
A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{200} + 3 x + 500 \) dollars, and each unit sells for \( \displaystyle \$63 \). Find the marginal cost at \( \displaystyle x = 10 \), and the number of units that maximizes profit, and that profit.
- \[ \left. \frac{d}{d x} \left(\frac{x^{2}}{200} + 3 x + 500\right) \right|_{\substack{ x=10 }} = \frac{31}{10} \]Marginal cost C′(10): about the cost of unit 11.✓ Proved
- \[ 63 x - \left(\frac{x^{2}}{200} + 3 x + 500\right) = - \frac{x^{2}}{200} + 60 x - 500 \]Profit = revenue − cost.✓ Proved
- \[ \frac{d}{d x} \left(- \frac{x^{2}}{200} + 60 x - 500\right) = 60 - \frac{x}{100} \]P′(x).✓ Proved
- \[ \left. 60 - \frac{x}{100} \right|_{\substack{ x=6000 }} = 0 \]P′ = 0 at x = 6000.✓ Proved
- \[ \frac{d^{2}}{d x^{2}} \left(- \frac{x^{2}}{200} + 60 x - 500\right) = - \frac{1}{100} \]P″ < 0: a maximum.✓ Proved
- \[ \left. 60 x + \frac{\left(-1\right) x^{2}}{200} - 500 \right|_{\substack{ x=6000 }} = 179500 \]The maximum profit.✓ Proved
Answer \( C'(10) = \frac{31}{10},\ x = 6000,\ P(6000) = 179500 \)
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
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17gpt-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 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), but the phrasing is imprecise and potentially confusing regarding indexing. More importantly, the solution fails to explicitly state that x must be non-negative or check boundary conditions, though for a downward parabola the critical point is sufficient. However, the primary issue is the misleading comment in line 1 which conflates the derivative's approximation with the specific unit cost definition in a way that might confuse students about whether it's the 10th or 11th unit. Actually, C'(x) approximates C(x+1)-C(x), so C'(10) approximates C(11)-C(10), which is the cost of the 11th unit. The comment says 'about the cost of unit 11'. This is technically acceptable in many calculus contexts. Let's look closer. Is there a real error? The problem asks for 'the number of units that maximizes profit'. The solution finds x=6000. Is x=6000 an integer? Yes. Is it valid? Yes. The profit calculation is correct. The marginal cost calculation is correct. The second derivative test is correct. The only potential issue is the comment in line 1. 'Marginal cost C'(10): about the cost of unit 11.' This is a standard interpretation. Is it misleading? Some texts say 'cost of the next unit'. 'Unit 11' is specific. If x=10, the next unit is the 11th. This seems fine. Let's re-read the prompt requirements. 'Check that... nothing would teach a student something false.' Is there anything false? No. Is there anything wrong with the setup? No. Wait, look at line 1 again. `Subs(Derivative(...), x, 10) = 31/10`. The comment says `-- Marginal cost C′(10): about the cost of unit 11.` This is a standard heuristic. Is it an error? No. Is it misleading? Maybe slightly, but not enough to fail. Let's look for other issues. Did the solution check if x=6000 is a maximum? Yes, line 5. Did it calculate the profit? Yes, line 6. Did it calculate marginal cost? Yes, line 1. The answer matches the stated answer. The solution is sound. Why would I fail it? Perhaps the comment in line 1 is considered 'style' or 'misleading' if the student thinks it's exact? It says 'about'. That's good. I will pass it.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.