Question

A ladder $10$ metres long rests with one end against a vertical wall, the other end on the floor. The lower end moves away from the wall at the rate of $2$ metres/minute. The rate at which the upper end falls when its base is $6$ metres away from the wall is.

• A. $3$ metres/min
• B. $\frac{2}{3}$ metres/min
• C. $\frac{3}{2}$ metres/min
• D. None of these

Answer 1

Answer: (C)

$\frac{3}{2}$ metres/min

Given $\frac{dx}{dt}=2$ meter/min, $\frac{dy}{dt}$ $=?$
In a right angled triangle, we have
$x^2+y^2=100$
$2x\frac{dx}{dt}+2y\frac{dy}{dt}=0$
At $x=6$, then $y=8$
$\Rightarrow 2\times 6\times 2+2\times 8\times \frac{dy}{dt}=0$
$\frac{dy}{dt}=-\frac{24}{16}=\frac{3}{2}$ meter/minute
${\left(\frac{dy}{dt}\right)}_{(6,8)}=-\frac{3}{2}$ meter/minute