Business applications: cost, revenue and profit
Problem 3.734 · medium
A company's cost to make \( \displaystyle x \) units is \( \displaystyle C(x) = \frac{x^{2}}{50} + 8 x + 1000 \) dollars, and each unit sells for \( \displaystyle \$28 \). Find the marginal cost at \( \displaystyle x = 100 \), and the number of units that maximizes profit, and that profit.
- \[ \left. \frac{d}{d x} \left(\frac{x^{2}}{50} + 8 x + 1000\right) \right|_{\substack{ x=100 }} = 12 \]Marginal cost C′(100): about the cost of unit 101.✓ Proved
- \[ 28 x - \left(\frac{x^{2}}{50} + 8 x + 1000\right) = - \frac{x^{2}}{50} + 20 x - 1000 \]Profit = revenue − cost.✓ Proved
- \[ \frac{d}{d x} \left(- \frac{x^{2}}{50} + 20 x - 1000\right) = 20 - \frac{x}{25} \]P′(x).✓ Proved
- \[ \left. 20 - \frac{x}{25} \right|_{\substack{ x=500 }} = 0 \]P′ = 0 at x = 500.✓ Proved
- \[ \frac{d^{2}}{d x^{2}} \left(- \frac{x^{2}}{50} + 20 x - 1000\right) = - \frac{1}{25} \]P″ < 0: a maximum.✓ Proved
- \[ \left. 20 x + \frac{\left(-1\right) x^{2}}{50} - 1000 \right|_{\substack{ x=500 }} = 4000 \]The maximum profit.✓ Proved
Answer \( C'(100) = 12,\ x = 500,\ P(500) = 4000 \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation. 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 — The solution correctly sets up 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, optimal units, and maximum profit.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly sets up 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, optimal units, and maximum profit.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-18gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly models the problem, applies calculus rules accurately, and interprets the results correctly. The marginal cost calculation, profit function derivation, optimization, and second derivative test are all sound.gpt-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.