Home›Calculus 3›Quadric surfaces in standard form›Problem 9.727

Quadric surfaces in standard form

Problem 9.727 · easy

Write \( \displaystyle 18 x^{2} + 32 y^{2} + 72 z^{2} = 288 \) in standard form and identify the surface.

Divide through by 288.
\[ \frac{18 x^{2} + 32 y^{2} + 72 z^{2}}{288} = \frac{x^{2}}{16} + \frac{y^{2}}{9} + \frac{z^{2}}{4} \]
The standard form.✓ Proved
A ellipsoid: x²/a² + y²/b² + z²/c² = 1.

Answer \( \frac{x^{2}}{16} + \frac{y^{2}}{9} + \frac{z^{2}}{4} = 1\ (\text{ellipsoid}) \)

Lines: 1 proved, 2 not checked. 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	Not checked	—	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	Not checked	—	a sentence; read, not computed
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	the surface classified from the signs of its coefficients; the standard form is a multiple of the given equation

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (style) — The article 'A' is incorrect before 'ellipsoid'; it should be 'An ellipsoid'. While the math is correct, the grammar error is distracting.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-10-18 — The article 'A' is incorrect before 'ellipsoid'; it should be 'An ellipsoid'. While the math is correct, the grammar error is distracting.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly divides by the constant term to obtain the standard form and correctly identifies the surface as an ellipsoid.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (style) 2026-10-15 — The article 'A' should be 'An' before 'ellipsoid'. Additionally, step 2 is marked as unchecked but contains the result of the division; while the algebra is correct, the label is inconsistent with the prompt's instructions about checked equations.
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/quadric_surfaces, checked 2026-10-18 with SymPy 1.14.0.