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
(1) $\frac{9}{7}$
(2) $\frac{7}{9}$
(3) $\frac{2}{9}$
(4) $\frac{9}{2}$

A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is $\tan ^ { - 1 } \left( \frac { 1 } { 2 } \right)$. Water is poured into it at a constant rate of $5$ cubic $\mathrm { m } / \mathrm { min }$. Then the rate (in $\mathrm { m } / \mathrm { min }$), at which the level of water is rising at the instant when the depth of water in the tank is $10 m$; is:
(1) $\frac { 1 } { 10 \pi }$
(2) $\frac { 1 } { 15 \pi }$
(3) $\frac { 1 } { 5 \pi }$
(4) $\frac { 2 } { \pi }$

A spherical iron ball of 10 cm radius is coated with a layer of ice of uniform thickness that melts at a rate of $50 \mathrm {~cm} ^ { 3 } / \mathrm { min }$. When the thickness of ice is 5 cm , then the rate (in $\mathrm { cm } / \mathrm { min }$.) at which of the thickness of ice decreases, is:
(1) $\frac { 5 } { 6 \pi }$
(2) $\frac { 1 } { 54 \pi }$
(3) $\frac { 1 } { 36 \pi }$
(4) $\frac { 1 } { 18 \pi }$

A particle is moving in a straight line such that its velocity is increasing at $5 \mathrm{~m}\mathrm{~s}^{-1}$ per meter. The acceleration of the particle is $\_\_\_\_$ $\mathrm{m}\mathrm{~s}^{-2}$ at a point where its velocity is $20 \mathrm{~m}\mathrm{~s}^{-1}$.

Water is being filled at the rate of $1 \mathrm{~cm}^3 \mathrm{sec}^{-1}$ 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 $\mathrm{cm}^2 \mathrm{sec}^{-1}$) at which the wet conical surface area of the vessel increases is
(1) 5
(2) $\frac{\sqrt{21}}{5}$
(3) $\frac{\sqrt{26}}{5}$
(4) $\frac{\sqrt{26}}{10}$

A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of $81 \mathrm {~cm} ^ { 3 } / \mathrm { min }$ and the thickness of the ice-cream layer decreases at the rate of $\frac { 1 } { 4 \pi } \mathrm {~cm} / \mathrm { min }$. The surface area (in $\mathrm { cm } ^ { 2 }$) of the chocolate ball (without the ice-cream layer) is :
(1) $196 \pi$
(2) $256 \pi$
(3) $225 \pi$
(4) $128 \pi$