Answer
$3k$
where k is a constant of proportionality
Work Step by Step
Let volume of the cone=V
Radius of the cone=r
height of the cone=h
Then
$V=\frac{1}{3}\pi r^2 h$ ....................... eq (1)
$\frac{dV}{dt}=$ Rate of evaporation of water
If r is the radius at any instant during evaporation ,then surface area $=\pi r^2$
According to the given condition
$\frac{dV}{dt} \propto \pi r^2 $ Or
$\frac{dV}{dt} =k \pi r^2 $ ............................ eq (2)
Where k is a constant of proportionality
Putting equation (1) in equation (2)
$\frac{d (\frac{1}{3}\pi r^2 h)}{dt} =k \pi r^2 $ .
$\frac{d (\frac{1}{3}\pi r^2 h)}{dt} =k \pi r^2 $ .
Considering r as constant, taking derivative with respect to height h only
$ \frac{1}{3} \pi r^2 \frac{d h}{dt} =k \pi r^2 $ .Or
$\frac{dh}{dt}=3k$
The required result
