$A$ spherical balloon is filled with $4500\pi$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72\pi$ cubic meters per minute,then the rate (in meters per minute) at which the radius of the balloon decreases $49$ minutes after the leakage began is:

Water is being filled at the rate of $1 \, cm^3/sec$ in a right circular conical vessel (vertex downwards) of height $35 \, cm$ and diameter $14 \, cm$. When the height of the water level is $10 \, cm$,the rate (in $cm^2/sec$) at which the wet conical surface area of the vessel increases is

Let $x$ be the length of one of the equal sides of an isosceles triangle,and let $\theta$ be the angle between them. If $x$ is increasing at the rate of $1/12 \ m/hr$,and $\theta$ is increasing at the rate of $\pi/180 \ \text{radians/hr}$,then find the rate in $m^2/hr$ at which the area of the triangle is increasing when $x = 12 \ m$ and $\theta = \pi/4$.

The rate of change of the volume of a sphere with respect to its surface area when the radius is $5 \ m$ is:

$A$ water tank has the shape of an inverted right circular cone,whose semi-vertical angle is $\tan^{-1}(1/2)$. Water is poured in at a constant rate of $5 \ m^3/min$. Then the rate (in $m/min$) at which the level of water is rising at the instant when the depth of water in the tank is $10 \ m$ is:

The radius of a right circular cylinder increases at the rate of $0.1 \text{ cm/min}$ and the height decreases at the rate of $0.2 \text{ cm/min}$. The rate of change of the volume of the cylinder in $\text{cm}^3\text{/min}$,when the radius is $2 \text{ cm}$ and the height is $3 \text{ cm}$,is: