Volume and Surface Area of Different Shapes

Q. If a right circular cone having maximum volume is inscribed in a sphere of radius $3$ cm, then the curved surface area (in cm${}^2$) of this cone is :

1. $6\sqrt{2}\pi$
2. $6\sqrt{3}\pi$
3. $8\sqrt{2}\pi$
4. $8\sqrt{3}\pi$

Q. A cylinder is inscribed in a sphere of radius $3$ units. Find the curved surface area of the cylinder which has the maximum volume.

1. $6\sqrt{2}\pi$
2. $12\sqrt{2}\pi$
3. $8\sqrt{3}\pi$
4. $6\sqrt{3}\pi$

Q. A spherical balloon is being inflated so that its volume increases uniformly at the rate of $40{\text{cm}}^3\text{/min.}$

At radius $r=8\text{cm}$, its surface area increases at the rate

1. $8{\text{cm}}^2\text{/min}$
2. $10{\text{cm}}^2\text{/min}$
3. $20{\text{cm}}^2\text{/min}$
4. None of these

Q. The height of a right circular cylinder of maximum volume inscribed in a sphere of radius of $3$ is :

1. $\frac{2}{3}\sqrt{3}$
2. $2\sqrt{3}$
3. $\sqrt{6}$
4. $\sqrt{3}$

Q. 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 meter per minute. 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 $10m$, is :

Q. Water is filled into a right inverted conical tank at a constant rate of $3{\text{m}}^3\text{/sec},$ whose semi vertical angle is ${\cos }^{-1}\frac{4}{5}.$ The rate $\left(\text{in}\text{m}\text{/sec}\right),$ at which the level of water is rising at the instant when the depth of water in the tank is $4\text{m},$ is

1. $\frac{1}{\pi }$
2. $\frac{1}{2\pi }$
3. $\frac{1}{3\pi }$
4. $\frac{1}{4\pi }$

Q. A wire of length 2 units is cut into two parts which are bent respectively to form a square of side = x units and a circle of radius = r units. If the sum of the areas of the square and the circle so formed is minimum, then:

1. $x=2r$
2. $2x=r$
3. $2x=(\pi +4)r$
4. $(4-\pi )x=\pi r$

Q.

Let (h, k) be a fixed point, where h > 0, k > 0. A straight line passing through this point cuts the positive direction of the coordinate axes at the points P and Q. The minimum area of the $\Delta$ OPQ, O being the origin, is

1. 2hk

2. kh

3. 4kh

4. 3hk