Question

A ladder $5m$ long is learning against a wall. The foot of the ladder is pulled out along the ground away from the wall at a rate $2m/s$. How fast is the height of ladder on the wall decreasing at the instant when the foot of the ladder is $4m$ away from the wall

• A. $10m/s$
• B. $\frac{3}{2}m/s$
• C. $\frac{8}{3}m/s$
• D. $\frac{1}{10}m/s$

Answer 1

The correct option is C $\frac{8}{3}m/s$

Given,

The length of the ladder is$5m$

$\frac{dx}{dt}=2cm/s$

The distance of the foot, $x=4m$ Height of the wall $=y=\sqrt{5^2-4^2}=3m$ (By Pythagoras theorem)

When, $x=5,y=3$ Also, $x^2+y^2=5^2$

Differentiate w.r.t then we get,

$2x\frac{dx}{dt}+2y\frac{dy}{dt}=0\Rightarrow x\frac{dx}{dt}+y\frac{dy}{dt}=0$

Now put the value of x and y then we get,

$4\times 2+3\times \frac{dy}{dt}=0\Rightarrow \frac{dy}{dt}=\frac{-8}{3}cm/s$