Home›Calculus 1›Related rates›Problem 3.2

Related rates

Problem 3.2 · easy

A 10 ft ladder leans against a vertical wall. The bottom slides away from the wall at 1 ft/s. How fast is the top sliding down the wall when the bottom is 8 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
\[ 6 \]
When x = 8, y = √(100 − 64).Reviewed
\[ \left. \frac{- v x}{y} \right|_{\substack{ x=8\\ y=6\\ v=1 }} = - \frac{4}{3} \]
dy/dt = −(x/y)(dx/dt) with x = 8, y = 6, dx/dt = 1.✓ Proved

Answer \( - \frac{4}{3} \ \text{ft/s} \approx -1.333\ \text{ft/s} \)

✓ 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.

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.
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 implicit differentiation, calculates the missing side length, and substitutes the values accurately to find the rate.
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.