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 $\frac{8}{3}$

since ABC is a right angled triangle
$A{C}^2+A{B}^2=B{C}^2\Rightarrow x^2+y^2={20}^2..............................\left(1\right)$
when it is 12 ft away from wall (means $x=12$ )
$A{C}^2+{12}^2={20}^2\Rightarrow AC=16\Rightarrow y=16$
Differentiating (1) w.r.t time $t$
$\Rightarrow 2x\frac{dx}{dt}+2y\frac{dy}{dt}=0$
Putting $x=12,y=16,\frac{dx}{dt}=v,$ $\frac{dy}{dt}=-2{(}^{\prime }{-}^{\prime }becauseyisdecreasing)$ we get
$\Rightarrow 2\times 12\times v-2\times 16\times 2=0\Rightarrow v=\frac{8}{3}$
Therefore Answer is $A$