Question

A ladder $20$ ft long leans against a vertical wall. The top end slides downwards at the rate of $2$ ft per second. The rate at which the lower end moves on a horizontal floor when it is $12$ ft from the wall is

• A. $\frac{8}{3}$
• B. $\frac{6}{5}$
• C. $\frac{3}{2}$
• D. $\frac{17}{4}$

Answer 1

The correct option is A $\frac{8}{3}$

From figure, $x^2+y^2=400$
Given $\frac{dy}{dt}=2\text{ft/sec}$

$x^2+y^2=400$
Differentiate w.r.t $t$,
$2x\frac{dx}{dt}+2y\frac{dy}{dt}=0$
$\Rightarrow x\frac{dx}{dt}=-2y$

Now, when $x=12\Rightarrow y=16$
$\therefore 12\times \frac{dx}{dt}=-2\times 16$
$\Rightarrow \frac{dx}{dt}=-\frac{8}{3}$