Conversion of solid from one shape to another

Q.

From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter as that of cylinder is hollowed out. Find the total surface area of the remaining solid. $(Use\pi =\frac{22}{7})$

1. $19.60{\text{cm}}^2$

2. $18.60{\text{cm}}^2$

3. $17.60{\text{cm}}^2$

4. $15.60{\text{cm}}^2$

Q.

Question 15
A building is in the form of a cylinder surmounted by a hemispherical vaulted dome and contains $41\frac{19}{21}{m}^3$ of air. If the internal diameter of dome is equal to its total height above the floor, find the height of the building?

Q.

$2.2d{m}^3$ of brass is to be drawn into a cylindrical wire of diameter $0.50cm$. Find the length of the wire.

Q.

The height of a cone is 40 cm. A small cone is cut off at the top by a plane parallel to the base. If the volume of the small cone be $\frac{1}{64}$ of the volume of the given cone, at what height ( in cm) above the base is the section made ?

1. 20

2. 30

3. 40

4. 50

Q.

Question 12
A milk container of height 16 cm is made of metal sheet in the form of a frustum of a cone with radii of its lower and upper ends as 8 cm and 20 cm respectively. Find the cost of milk at the rate of ₹ 22 per litre which the container can hold.

Q. Question 18
The diameters of the two circles ends of the bucket are 44 cm and 24 cm. The height of the bucket is 35 cm. The capacity of the bucket is
(A) 32.7 litres
(B) 33.7 litres
(C) 34.7 litres
(D) 31.7 litres

Q.

If the ratio of the radius of a cone and a cylinder of equal volume is 3:5, then find the ratio of their heights.

1. $\frac{25}{3}$

2. $\frac{28}{3}$

3. $\frac{23}{3}$

4. $7$

Q.

The outer and inner diameters of a hemispherical bowl are $17cm$ and $15cm$ respectively. Find the cost of polishing it all over at $25$ paise per $c{m}^2$. ( Take $\pi$ $=\frac{22}{7}$).

Q.

A drinking glass is in the shape of a frustum of a cone of height 14 cm. The diameter of its two circular ends are 4 cm and 2 cm. The capacity of the glass is $102\frac{2}{3}c{m}^3$.

1. 

2. 

Q.

A hollow cylindrical pipe is $210cm$ long. Its outer and inner diameters are $10cm$ and $6cm$ respectively. Find the volume of the copper used in making the pipe ___

Q.

The volume of the largest sphere than can be cut from a cylindrical log of wood of base radius 1 m and height 4 m is:

1. 

2. 

3. 

4. 

Q.

Water is leaking out of an inverted conical tank at a rate of $10,000{\text{cm}}^{\text{3}}\text{/min}$ at the same time that water is being pumped into the tank at a constant rate.

The tank has a height of $6\text{m}$, and the diameter at the top is $4\text{m}$.

If the water level is rising at a rate of $20\text{cm/min}$ when the height of the water is $2\text{m}$, find the rate at which water is being pumped into the tank.

(Note: $\frac{dV}{dt}=\left(\text{rate of pumped in water}\right)-\left(\text{rate of leak}\right)$ ).

Q.

Question 6
Derive the formula for the curved surface area and total surface area of the frustum of cone.

Q. A girl fills a cylindrical bucket 32 cm in height and 18 cm in radius with sand. She empties the bucket on the ground and makes a conical heap of the sand. If the height of the conical heap is 24 cm, find:
(i) its radius and
(ii) Its slant height. (Leave your answer in square root form) [4 MARKS]

Q.

A container made up of a metal sheet is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm respectively. Find the cost of the milk which can completely fill the container at the rate of Rs. 15 per litre and the cost of the metal sheet used, if it costs Rs. 5 per 100 $c{m}^2$. (Take $\pi$ = 3·14)

1. 95.63

2. 100

3. 97.97

4. 92.45

Q.

From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter as that of cylinder is hollowed out. Find the total surface area of the remaining solid. $(Use\pi =\frac{22}{7})$

1. $19.60{\text{cm}}^2$

2. $18.60{\text{cm}}^2$

3. $17.60{\text{cm}}^2$

4. $15.60{\text{cm}}^2$

Q.

Question 5
A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0.5 cm are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.

Q.

A rectangular paper is folded into a cylinder. The length and breadth of the paper are L and B respectively. What is the outer surface area of the cylinder?

1. LB

2. $\pi$LB

3. 2$\pi$LB

4. 2LB

Q.

A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0.5 cm are dropped into the vessel, one-fourth of the water flows out. The number of lead shots dropped in the vessel is 100.

1. 110

2. 100

Q. Question 4
A cone of radius 8 cm and height 12 cm is divided into two parts by a plane through the mid-point of its axis parallel to its base. Find the ratio of the volumes of two parts

Q.

The difference between the outside and inside surface of a cylindrical metallic pipe 14 cm long is $44c{m}^2$. If the pipe is made of 99 $c{m}^3$ of metal, find the outer and inner radii of the pipe. [4 MARKS]

Q.

A Ramraj pencil box of dimensions 30cm × 5cm × 5cm consists of 10 sharpened cylindrical pencils placed in a vertical fashion each of length 20cm and diameter 1.4cm and having a conical surmount whose slant height is 2.5m. There are 2 rows of 5 pencils each. Find available volume in the box. Can a sharpener and an eraser in shape of a cube each of side 5cm be placed on top of the pencils such that sharpener is placed on top of the eraser.

1. 426.5 , Yes

2. 426.8 , Yes

3. 429.8 , No

4. 426.5 , No

Q.

There is a toy rocket in the shape shown below. The base radius of the cylinder = 1.5 cm. Base radius of the cone = 2.5 cm. Find the total surface area (in sq.cm) of the toy.

1. 60$\pi$

2. 70$\pi$

3. 90$\pi$

4. 82.5$\pi$

Q. Question 1
Write ‘True’ or ‘False’ and justify your answer in the following :
Two identical solid hemispheres of equal base radius r cm are stuck together along their bases. The total surface area of the combination $6\pi r^2$.

Q.

Volume enclosed between a cylindrical vessel consisting of 2 concentric cylinders is 90$\pi$ $m^3$. If outer radius is 5m and height of the cylinder is 10m. Find the thickness of the cylindrical vessel.

1. 2.5m

2. 1m

3. 1.5m

4. 2m

Q. (a) A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top which is open is 5 cm. It is filled with water up to the brim. When spherical lead shots, each of radius 0.5 cm, are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.

(b) A certain number of metallic cones, each of radius 2 cm and height 3 cm are melted and recast into a solid sphere of radius 6 cm. Find the number of cones. [6 MARKS]

Q.

If Hemakshi reshaped a cone of height h cm and radius of base r cm into a sphere, then, which of the following options is always correct?

1. Volume of cone = Volume of sphere

2. Surface area of cone = Surface area of sphere

3. Radius of cone = Radius of sphere

4. None of the above

Q.

2.2 cu dm(decimeter cube) of brass is to be drawn into a cylindrical wire of diameter 0.50 cm. Find the length of the wire(in m).

___

Q. The number of ways in which 8 boys be seated at a round table so that two particular boys are next to each other is

1. 7!2!
2. 8!2!
3. 6!2!
4. 6!

Q.

2.2 cu dm(decimeter cube) of brass is to be drawn into a cylindrical wire of diameter 0.50 cm. Find the length of the wire. [1 MARK]