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Related rates

Problem 3.7 · easy

A 10 ft ladder leans against a vertical wall. The bottom slides away from the wall at 4 ft/s. How fast is the top sliding down the wall when the bottom is 6 ft from the wall?

Let x be the distance from the wall to the bottom and y the height of the top. The ladder is the hypotenuse: x² + y² = 100.
Reviewed
Differentiate with respect to t: 2x dx/dt + 2y dy/dt = 0, so dy/dt = -(x/y) dx/dt.
Reviewed
\[ 8 \]
When x = 6, y = √(100 − 36).Reviewed
\[ \left. \frac{- v x}{y} \right|_{\substack{ x=6\\ y=8\\ v=4 }} = -3 \]
dy/dt = −(x/y)(dx/dt) with x = 6, y = 8, dx/dt = 4.✓ Proved

Answer \( -3 \ \text{ft/s} \approx -3\ \text{ft/s} \)

✓ Nihil obstat Lines: 1 proved, 3 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
3	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
4	✓ 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	the situation was stepped forward and back by a microsecond and the quantity differenced numerically

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly sets up the related rates problem, applies implicit differentiation, and substitutes the correct values to find the rate of change of the height.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly sets up the related rates problem, applies implicit differentiation, and substitutes the correct values to find the rate of change of the height.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly sets up the related rates problem, applies the chain rule properly, and substitutes the correct values to find the rate of change.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly sets up the related rates problem, applies implicit differentiation, and substitutes the correct values to find the rate of change.
gpt-oss:20b: pass 2026-09-26

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/related_rates, checked 2026-10-13 with SymPy 1.14.0.