Optimization
Problem 3.90 · medium
Equal squares are cut from the corners of a 8 in by 8 in sheet of cardboard and the sides folded up to make an open box. What cut maximizes the volume, and what is the volume?
- Cutting squares of side x leaves a base 8 − 2x on a side and height x: V(x) = x(8 − 2x)², 0 < x < 4.
- \[ \frac{d}{d x} x \left(8 - 2 x\right)^{2} = \left(3 x - 4\right) \left(4 x - 16\right) \]V'(x), factored.✓ Proved
- \[ \left. x \left(8 x - 32\right) + \left(8 - 2 x\right)^{2} \right|_{\substack{ x=\frac{4}{3} }} = 0 \]V'(x) = 0 at x = 4/3 (the other root, x = 4, gives no box).✓ Proved
- \[ \left. x \left(8 - 2 x\right)^{2} \right|_{\substack{ x=\frac{4}{3} }} = \frac{1024}{27} \]The maximum volume.✓ Proved
- V is 0 at both ends of the interval and positive in between, so the interior critical point is the maximum.
Answer \( \frac{1024}{27} \)
Lines: 3 proved, 2 not checked. The answer was also checked a second way, without looking at the solution. The explanation has not been reviewed yet.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 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 | Not checked | — | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the objective sampled at 200,001 points of its interval tops out at the same value |
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/optimization, checked 2026-09-26 with SymPy 1.14.0.