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Equations of spheres

Problem 9.841 · easy

Find the equation of the sphere with center \( \displaystyle (-3, 0, 5) \) and radius \( \displaystyle 4 \), in standard form.

A sphere is every point at distance r from the center: (x − h)² + (y − k)² + (z − l)² = r².
\[ \left. y^{2} + \left(x + 3\right)^{2} + \left(z - 5\right)^{2} \right|_{\substack{ x=1\\ y=0\\ z=5 }} = 16 \]
A point at distance r along the x-direction is on it.✓ Proved

Answer \( y^{2} + \left(x + 3\right)^{2} + \left(z - 5\right)^{2} = 16 \)

Lines: 1 proved, 1 not checked. 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	Not checked	—	a sentence; read, not computed
2	✓ 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	points at the claimed radius from the claimed center satisfy the equation

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly state the final equation of the sphere, which is the answer to the problem. It only provides a check that a specific point lies on the sphere, but does not present the derived equation as the conclusion.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-20 — The solution fails to explicitly state the final equation of the sphere, which is the answer to the problem. It only provides a check that a specific point lies on the sphere, but does not present the derived equation as the conclusion.
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the standard equation of a sphere and verifies it with a valid point on the sphere. The setup and conclusion are sound.
gpt-oss:20b: fail (misleading) 2026-10-19 — The solution claims to prove the sphere equation by checking a single point, which does not establish the general equation. The derivation is incomplete and could mislead a student into thinking one point suffices to verify the sphere’s equation.
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the standard equation of a sphere and substitutes the given center and radius. The verification step confirms a point on the sphere satisfies the equation.
gpt-oss:20b: pass 2026-10-19

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/sphere_equation, checked 2026-10-20 with SymPy 1.14.0.