Question

A ladder 10 m long rests against a vertical wall with the lower end on the horizontal ground. The lower end of the ladder is pulled along the ground away from the wall at the rate of $3cm/s$ The height of the upper end while it is descending at the rate of $4cm/s$, is

• A. $4\sqrt{3}m$
• B. $5\sqrt{3}m$
• C. $5\sqrt{2}m$
• D. $6m$

Answer 1

The correct option is D $6m$

Let $AB=x,BC=y$ and $AC=10m$

$\therefore x^2+y^2=100$ ....... $\left(i\right)$

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

$\Rightarrow 2x\left(3\right)-2y\left(4\right)=0$ $[\because \frac{dx}{dt}=3cm/s,\frac{dy}{dt}=-4cm/s]$

$\Rightarrow 6x-8y=0$

$\Rightarrow x=\frac{4y}{3}$

On putting this value in Eq. $\left(i\right)$, we get

$\frac{16}{9}{y}^2+y^2=100$

$\Rightarrow \frac{25y^2}{9}=100$

$\Rightarrow y^2=36$

$\Rightarrow y=6$

Hence, the height of the upper end is $6m$.