Volume and Surface Area of Different Shapes

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. 
3. 
4. None of these

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. 4kh

3. kh

4. None of these

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. A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of V $m{m}^3$, has a 2mm thick solid wall and is open at the top. The bottom of the container is a solid circular disc of thickness 2mm and is of radius equal to the outer radius of the container.
If the volume of the material used the container is minimum when the inner radius of the container is 10mm, then the value of $\frac{V}{250\pi }$ is ___

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

When radius $r=8\text{cm}$, then the increase in radius in the next $1/2\text{min}$ is

1. $0.025\text{cm}$
2. $0.01\text{cm}$
3. $0.075\text{cm}$
4. $0.050\text{cm}$

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

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 :

1. $\frac{1}{15\pi }$
2. $\frac{2}{\pi }$
3. $\frac{1}{5\pi }$
4. $\frac{1}{10\pi }$

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. 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. $6\sqrt{3}\pi$