Home›Calculus 3›Equations of spheres›Problem 9.698

Equations of spheres

Problem 9.698 · easy

Find the equation of the sphere with center \( \displaystyle (2, -3, 3) \) and radius \( \displaystyle 3 \), in standard form.

A sphere is every point at distance r from the center: (x − h)² + (y − k)² + (z − l)² = r².
Reviewed
\[ \left. \left(x - 2\right)^{2} + \left(y + 3\right)^{2} + \left(z - 3\right)^{2} \right|_{\substack{ x=5\\ y=-3\\ z=3 }} = 9 \]
A point at distance r along the x-direction is on it.✓ Proved

Answer \( \left(x - 2\right)^{2} + \left(y + 3\right)^{2} + \left(z - 3\right)^{2} = 9 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — 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.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — 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: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — 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 lies at the correct distance.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the standard equation of a sphere and substitutes the given center and radius. The unchecked equation in step 2 is a valid verification step (checking a point on the sphere) and does not contain errors.
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/sphere_equation, checked 2026-10-18 with SymPy 1.14.0.