Volume and Surface Area of Different Shapes

Q. The volume of the greatest cylinder which can be inscribed in a cone of height $30$ cm and semi-vertical angle ${30}^{\circ }$ is

1. $4000\frac{\pi }{3}$ cubic cm
2. $400\frac{\pi }{3}$ cubic cm
3. $4000\frac{\pi }{\sqrt{3}}$ cubic cm
4. None of these

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. The maximum volume (in cu.m) of the right circular cone having slant height 3 m is:

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

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

Q. A box, constructed from a rectangular metal sheet, is $21$ cm by $16$ cm by cutting equal squares of side $x$ cm from the corners of the sheet and then turning up the projected portions. The value of $x$ (in cm) so that volume of the box is maximum is

1. $1$
2. $2$
3. $3$
4. $4$

Q. Which of the following is/are true for the right circular cone with maximum volume (in cu.m) and having slant height $3m$

1. 
   $h=\sqrt{6}$
2. $h=\sqrt{3}$
   
3. $V_{max}=2\sqrt{3}\pi$
4. $V_{max}=\sqrt{6}\pi$
   

Q. The coordinates of the point P(x, y) lying in the first quadrant on the ellipse $\frac{x^2}{8}+\frac{y^2}{18}=1$ so that the area of the triangle formed by the tangent at P and the coordinate axes is the smallest, are given by

1. (2, 3)
2. $(\sqrt{8},0)$
3. $(\sqrt{18},0)$
4. (3, 2)

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

Q. An open cylindrical can has to be made with $100$ square units of tin. If its volume is maximum, then the ratio of its base radius and the height is

1. $2:1$
2. $1:1$
3. $1:2$
4. $\sqrt{2}:1$