Business applications: cost, revenue and profit
Problem 3.718 · medium
A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{20} + 3 x + 500 \) dollars, and each unit sells for \( \displaystyle \$43 \). Find the marginal cost at \( \displaystyle x = 10 \), and the number of units that maximizes profit, and that profit.
- \[ \left. \frac{d}{d x} \left(\frac{x^{2}}{20} + 3 x + 500\right) \right|_{\substack{ x=10 }} = 4 \]Marginal cost C′(10): about the cost of unit 11.✓ Proved
- \[ 43 x - \left(\frac{x^{2}}{20} + 3 x + 500\right) = - \frac{x^{2}}{20} + 40 x - 500 \]Profit = revenue − cost.✓ Proved
- \[ \frac{d}{d x} \left(- \frac{x^{2}}{20} + 40 x - 500\right) = 40 - \frac{x}{10} \]P′(x).✓ Proved
- \[ \left. 40 - \frac{x}{10} \right|_{\substack{ x=400 }} = 0 \]P′ = 0 at x = 400.✓ Proved
- \[ \frac{d^{2}}{d x^{2}} \left(- \frac{x^{2}}{20} + 40 x - 500\right) = - \frac{1}{10} \]P″ < 0: a maximum.✓ Proved
- \[ \left. 40 x + \frac{\left(-1\right) x^{2}}{20} - 500 \right|_{\substack{ x=400 }} = 7500 \]The maximum profit.✓ Proved
Answer \( C'(10) = 4,\ x = 400,\ P(400) = 7500 \)
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 2026-10-18gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (misleading) 2026-10-15 — The comment on line 1 claiming marginal cost is 'about the cost of unit 11' is misleading; marginal cost C'(x) approximates the cost of the (x+1)-th unit, but for x=10, it approximates the 11th unit's cost, which is technically correct in approximation but often confused with the exact cost. More importantly, the phrasing 'about the cost of unit 11' suggests C'(10) ≈ C(11) - C(10), which is a valid approximation, but the standard definition is the instantaneous rate of change. However, the bigger issue is that the problem asks for 'the number of units', implying an integer, but the solution treats x as a continuous variable without justification or rounding discussion, which is standard in calculus but can be misleading if not noted. Actually, the main error is likely the interpretation of marginal cost. C'(10) = 4. The cost of the 11th unit is C(11)-C(10) = (121/20 + 33 + 500) - (100/20 + 30 + 500) = (6.05 + 33) - (5 + 30) = 39.05 - 35 = 4.05. So 4 is indeed 'about' 4.05. This is acceptable. Let's look closer. Is there a calculation error? C'(x) = x/10 + 3. C'(10) = 1 + 3 = 4. Correct. P(x) = 43x - x^2/20 - 3x - 500 = -x^2/20 + 40x - 500. Correct. P'(x) = -x/10 + 40. P'(400) = -40 + 40 = 0. Correct. P''(x) = -1/10 < 0. Correct. P(400) = -160000/20 + 16000 - 500 = -8000 + 16000 - 500 = 7500. Correct. The solution is mathematically sound. Why would it fail? Maybe the 'about the cost of unit 11' is considered too informal or potentially confusing? Or perhaps the prompt implies discrete units? 'Find the number of units' usually implies an integer answer in business contexts, but in calculus problems, we often accept the continuous optimum. If x must be an integer, 400 is an integer, so it's fine. The note on line 1 is slightly imprecise (marginal cost is the derivative, not the discrete difference), but 'about' covers that. I will mark it as pass unless there is a subtle error. Wait, looking at line 1 again: 'Marginal cost C′(10): about the cost of unit 11.' This is a standard interpretation. I see no error.gpt-oss:20b: pass 2026-10-15
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.