Question

A ladder 24 ft long leans against a vertical wall. The lower end is moving away at the rate of 3 ft\sec. Find the rate at which the top of the ladder is moving downwards, if its foot is 8 ft from the wall.

Answer 1

We know that $x^2+y^2={24}^2$ ...... $\left(1\right)$

$⟹2x\frac{dx}{dt}+2y\frac{dy}{dt}=0$

$⟹x\frac{dx}{dt}+y\frac{dy}{dt}=0$

$⟹x\frac{dx}{dt}+y(-3)=0$

$⟹x\frac{dx}{dt}-3y=0$ ...... $\left(2\right)$

Given $x=8$ ft

$⟹8^2+y^2={24}^2$

$⟹y=22.62$

From $\left(2\right)$

$8\frac{dx}{dt}-3\left(22.62\right)=0$

$8\frac{dx}{dt}=67.88$

$⟹\frac{dx}{dt}=8.4$ ft\sec