A $10$ m ladder leans against a vertical wall. The foot of the ladder slides away from the wall at $0.5$ m/s. Find the speed at which the top of the ladder descends when the foot is $7$ m from the wall.

medium 2 marks Class 12 · Mathematics · Application of Derivatives

Answer

- 0.49

m/s

The correct answer is $- 0.49$ m/s — Answer: $- 0.49$ m/s.

Step-by-step solution

Let $x$ be the horizontal distance and $y$ be the vertical height. The relation $x^{2} + y^{2} = 1 0^{2}$ holds.
Differentiate implicitly: $2 x, d x / d t + 2 y, d y / d t = 0$ .
Rearrange to get $d y / d t = - (x / y), d x / d t$ .
Substitute $x = 7$ , $d x / d t = 0.5$ and $y = 1 0^{2} - 7^{2} = 51 a pp r o x 7.14$ .
Compute $d y / d t = - (7/7.14) (0.5) a pp r o x - 0.49$ m/s.
Negative sign indicates the top is moving downward. The speed of descent is $0.49$ m/s.
Answer: $- 0.49$ m/s.

Common mistake: Forgetting to use the negative sign or misapplying the Pythagorean relation

Concept tested: Related rates

Published 12 September 2026 · Last updated 16 September 2026 · How this solution was prepared and checked