The side of an equilateral triangle is increasing at the rate of $2 \text{ cm/s}$. Find the rate at which the area is increasing when the side of the triangle is $20 \text{ cm}$.

Find the rate of change of the area of a circle with respect to its radius $r$ when $r=5 \text{ cm}$.

Moving along the $x$-axis are two points with $x = 10 + 6t$ and $x = 3 + t^2$. The speed with which they are receding from each other at the time of encounter is ........... $cm/sec$. ($x$ is in $cm$ and $t$ is in seconds)

$A$ water tank has the shape of a right circular cone with its axis vertical and vertex downwards. Its semi-vertical angle is $\tan^{-1} \frac{3}{4}$. Water is poured into it at a constant rate of $6 \text{ m}^3/\text{hr}$. The rate (in $\text{m}^2/\text{hr}$) at which the wet curved surface area of the tank is increasing when the depth of water in the tank is $4 \text{ m}$ is:

$A$ particle moves such that $x = 2 + 27t - t^3$. The direction of motion reverses after moving a distance of ... units.

Find the rate of change of the area of a circle with respect to its radius $r$ when $r=4 \, cm$. (in $\pi \, cm$)