Volume and Surface Area of Different Shapes

Q.

A spherical balloon is being inflated at the rate of $35cc/min$. The rate of increase of the surface area of the balloon, when its diameter is $14cm$, is

1. $7sqcm/min$

2. $10sqcm/min$

3. $17.5sqcm/min$

4. $28sqcm/min$

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.

Let ABCD be a quadrilateral with area 18, with side AB parallel to CD and AB = 2CD. Let AD be perpendicular to AB and CD. If a circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius is

1. 

2. 3

3. 2

4. 1

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. 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.

Show that the closed right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base.

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.

If at any instant $t$, for a sphere, $r$denotes the radius, $S$ denotes the surface area and $V$ denotes the volume, then what is $dV/dt$equal to?

1. $\frac{1}{2}S\frac{dr}{dt}$

2. $\frac{1}{2}r\frac{dS}{dt}$

3. $r\frac{dS}{dt}$

4. $\frac{1}{2}{r}^2\frac{dS}{dt}$

Q. If the rate of change of area of rhombus with respect to its side is equal to the side of the rhombus, then the angles of rhombus are

1. $\frac{\pi }{3}$ and $\frac{2\pi }{3}$
2. $\frac{\pi }{4}$ and $\frac{3\pi }{4}$
3. $\frac{\pi }{6}$ and $\frac{5\pi }{6}$
4. $\frac{5\pi }{12}$ and $\frac{7\pi }{12}$

Q.

A rectangular storage container with an open top is to have a volume of $10{\mathrm{m}}^3$.

The length of this base is twice the width.

Material for the base costs $\$5\mathrm{per}\mathrm{square}\mathrm{meter}$ .

Material for the sides costs $\$3\mathrm{per}\mathrm{square}\mathrm{meter}$ .

Find the cost of materials for the cheapest such container. (Round your answer to the nearest cent.)

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. $\sqrt{2}:1$

Q.

A water tank has the shape of an inverted right circular cone with its axis vertical and vertex lowermost. Its semi-vertical angle is ${\tan }^{-1}\left(0.5\right)$ . Water is poured into it at a constant rate of $5$ cubic meter per hour. Find the rate at which the level of the water is rising at the instant when the depth of water in the tank is $4m$.

Q. A sperical ball of salt is dissolving in water in such a manner that the rate of decrease of its volume at any instant is equal four times its surface area. The rate of decrease in its radius (in units/sec) will be

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

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

Q.

A well of diameter $3m$ is dug $14m$ deep the earth taken out of it has been spread evenly all around it in the shape of a circular ring of width $4m$to form an embankment. Find the height of the embankment

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. The radius of an air bubble is increasing at the rate of $\frac{1}{2}cm/s$. At what rate is the volume of the bubble increasing when the radius is $1cm$?

Q. A balloon, which always remains spherical, has a variable diameter
$\frac{3}{2}(2x+1).$
Find the rate of change of its volume with respect to $x$.

Q. A balloon, which always remains spherical on inflation, is being inflated by pumping in $900$ cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is $15cm$.

Q. Water is dropping out at the steady rate of $2cc/sec$ through a tiny hole at the vertex of a conical whose axis is vertical. When the slant height of the water is $4cm$ then the rate of decrease height of water is [Given that the vertex angle of the funnel is ${120}^{\circ }$]

1. $\frac{32}{27\pi }cm/sec$
2. $\frac{2}{3\pi }cm/sec$
3. $\frac{4}{3\pi }cm/sec$
4. $\frac{1}{3\pi }cm/sec$

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

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.

A thin wire has a length of $21.7cm$ and radius $0.46mm.$ Calculate the volume of the wire correct to two decimal places.

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

Q.

A balloon which remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the radius is 10 cm.

Q. A square piece of tin of side $12cm$ is to be made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum ? Also, find this maximum volume.

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.

The radius of the base and the height of a right circular cylinder are in the ratio of: $2:3$ and its volume is $1617cu.cm$. Find the total surface area of the cylinder. (Use $\mathrm{\pi }=\frac{22}{7}$​)

Q. Let the area of the triangle formed by $(1,2),(-3,0)$ and any point on the line $x-2y+k=0$ is $5$ sq. units. If possible values of $k$ are $k_1,k_2$ then locus of foot of the perpendicular drawn from $(k_1,0)$ to a variable line passing through $(0,k_2)$ is

1. $x^2+y^2=2x-8y$
2. $x^2+y^2+2x-8y=0$
3. $x^2+y^2+8x-2y=0$
4. $x^2+y^2=8x+2y$