The side of an equilateral triangle expands at the rate of $2 \, cm/s$. The rate of increase of its area when each side is $10 \, cm$ is:

$A$ man of height $2 \ m$ walks at a uniform speed of $5 \ km/hr$ away from a lamp post which is $6 \ m$ high. Find the rate at which the length of his shadow increases in $km/hr$.

$A$ vessel in the shape of an inverted cone of height $10 \ ft$ and semi-vertical angle $30^{\circ}$ is full of water. Due to a hole at the vertex, the slant height of the water in the vessel is decreasing at a constant rate of $\frac{1}{\sqrt{3}} \ ft/min$. The rate (in $cu. \ ft/min$) at which the volume of water in the vessel is decreasing, when the volume of water is $\frac{8 \pi}{\sqrt{3}} \ cu. \ ft$, is

$A$ ladder,$5 \ m$ long,rests against a vertical wall. If its top slides downwards at the rate of $10 \ cm/s$,then the angle between the ladder and the floor is decreasing at the rate of how many radians/second when its lower end is $4 \ m$ away from the wall?

Displacement $s$ of a particle at time $t$ is expressed as $s=2 t^3-9 t$. Find the acceleration at the time when the velocity vanishes.

The radius of a sphere is changing. At an instant of time, the rate of change in its volume and its surface area are equal. Then the value of the radius at that instant is?