Exercise 12.1
Multiple Choice Questions (Q. No. 1-10)
Question. 1 A cylindrical pencil sharpened at one edge is the combination of
(a) a cone and a cylinder
(b) frustum of a cone and a cylinder
(c) a hemisphere and a cylinder
(d) two cylinders
Solution. (a)
Explanation:
A cylindrical pencil sharpened at one edge is the combination of a cone and a cylinder as shown below:

Question. 2 A surahi is the combination of
(a) a sphere and a cylinder
(b) a hemisphere and a cylinder
(c) two hemispheres           
(d) a cylinder and a cone
Solution.              
Explanation:
The lower portion of a surahi is a sphere and its upper part (neck) is in cylindrical shape.

(a) a cone and a cylinder
(b) a hemisphere and a cone
(c) frustum of a cone and a cylinder
(d) sphere and cylinder
Solution. (b)
Explanation: Base of a plumbline is a cone and its top is a hemisphere.

(a) acone 
(b) frustum of a cone
(c) a cylinder                                
(d) a sphere
Solution. (b)
Explanation:
The shape of a glass (tumbler) is usually in the form of a frustum as the frustum of a cone is the shape obtained obtained by cutting a right circular cone by a plane parallel to the base of the cone.
Question. 5 The shape of a gilli, in the gilli-danda game (see figure) is a combination of

Question. 6 A shuttle cock used for playing badminton has the combination of
(a) a cylinder and a sphere                (b) a cylinder and a hemisphere
(c) a sphere and a cone      (d) frustum of a cone and a hemisphere
Solution. (d)
Explanation:
The shape of the shuttle cock is in the form of the frustum of a cone and a hemisphere.

Question. 7 A cone is cut through a plane parallel to its base and then the cone that is formed on one side of that
plane is removed. The new part that is left over on the other side of the plane is called
(a) a frustum of a cone     
(b) cone
(c) cylinder           
(d) sphere
Solution. (a)        
Explanation:
When a cone is through the plane parallel to its base then the part other than the cone is called the frustum of a cone.

Question. 8 If a hollow cube of internal edge 22cm is filled with spherical marbles of diameter 0.5cm and it is assumed that  space of the cube remains unfilled. Then, the number of marbles that the cube can accomodate is
(a) 142244           
(b) 142344
(c) 142444           
(d) 142544
Solution. (a)
Explanation:
Given, edge of the cube = 22cm
∴ Volume of the cube = (side)³ = (22)³ = 10648cm³     
Also, given diameter of marble = 0.5cm

Question. 9 A metallic spherical shell of internal and external diameters 4cm and 8cm, respectively is melted and
recast into the form a cone of base diameter 8cm. The height of the cone is
(a) 12cm               
(b) 14cm               
(c) 15cm               
(d) 18cm
Solution. (b)
Explanation:
Given, internal diameter of spherical shell = 4cm
And external diameter of shell = 8cm
∴ Internal radius of spherical shell, r₁= 4/2 = 2cm    
And external radius of shell, r₂= 8/2 = 4 cm


Question. 10 If a solid piece of iron in the form of a cuboid of dimensions 49cm × 33cm × 24cm, is moulded to form a solid sphere. Then, radius of the sphere is
(a) 21cm               
(b) 23cm               
(c) 25cm               
(d) 19cm
Solution. (a)
Explanation:
As a cuboid is moulded to form a sphere,

Hence, the radius of the sphere is 21cm.
Exercise 12.1
Multiple Choice Questions (Q. No. 11-20):
Question.11 A mason constructs a wall of dimensions 270cm × 300cm × 350cm with the bricks each of size 22.5cm × 11.25cm × 8.75cm and it is assumed that 1/8  space is covered by the mortar. Then, the number of bricks
used to construct the wall is
(a) 11100             
(b) 11200             
(c) 11000              
(d) 11 300
Solution. (b)
Explanation:
Volume of the wall = 270 × 300 × 350 = 28350000cm³
And, volume of one brick = 22.5 × 11.25 × 8.75 = 2214.84cm³
Given, volume of the wall covered by mortar = 1/8 part

Question.12 Twelve solid spheres of the same size are made by melting a solid metallic cylinder of base diameter 2cm and height 16cm. The diameter of each sphere is
(a) 4cm                                                                             
(b) 3cm                                                                             
(c) 2cm                                                                             
(d) 6cm
Solution. (c)         
Explanation:

Question. 13 The radii of the top and bottom of a bucket of slant height 45cm are 28cm and 7cm, respectively. The
curved surface area of the bucket is
(a) 4950cm²         
(b) 4951cm²              
(c) 4952cm²              
(d) 4953cm²
Solution. (a)
Explanation:
Given, the radius of top of the bucket, r₁ = 28cm
Radius of the bottom of the bucket, r₂ = 7cm
And, slant height of the bucket, l = 45cm
As a bucket is in the from of a frustum of a cone,

Question. 14 A medicine-capsule is in the shape of a cylinder of diameter 0.5cm with two hemispheres stuck to
each of its ends. The length of entire capsule is 2cm. The capacity of the capsule is
(a) 0.36cm³          
(b) 0 35cm³          
(c) 0.34cm³          
(d) 0.33cm³
Solution. (a)

Question. 15 If two solid hemispheres of same base radius r are joined together along their bases, then curved surface area of this new solid is
(a) 4πr² 
(b) 6πr² 
(c) 3πr²
(d) 8πr²
Solution. (a)
Explanation:
As two solid hemispheres of same base radius r are joined together along their bases then the new solid formed will be a sphere with the same radius r.
∴Curved surface area of the sphere = πr²
Question. 16 A right circular cylinder of radius r cm and height h cm (where, h > 2r)just encloses a sphere of Diameter
(a) r cm 
(b) 2r cm              
(c) h cm
(d) 2h cm
Solution. (b)
Explanation:
For a sphere just enclosed in a cylinder, its diameter will be equal to the diameter of cylinder whichis 2r cm.

Question. 17 During conversion of a solid from one shape to another, the volume of the new shape will
(a) increase          
(b) decrease
(c) remain unaltered           
(d) be doubled
Sol. (c)  
Explanation:
When a solid is converted from one shape to another, the volume of the new shape will remain unaltered (same).
Question. 18 The diameters of the two circular ends of the bucket are 44cm and 24cm. The height of the bucket is
35cm. The capacity of the bucket is
(a) 32.7 L             
(b) 33.7 L             
(c) 34.7 L              
(d) 31.7 L
Solution. (a)
Explanation:
Given, Height of the bucket, h = 35cm
And diameters of the two circular ends of the bucket are 44cm and 24cm.

Question. 19 In a right circular cone, the cross-section made by a plane parallel to the base is a
(a) circle
(b) frustum of a cone         
(c) sphere              
(d) hemisphere
Solution. (b)
Explanation. If a right circular cone is cut by a plane parallel to the base then the cross-section obtained is acircle.
Question. 20 If volumes of two spheres are in the ratio 64:27, then the ratio of their surface areas is
(a) 3:4   
(b) 4:3   
(c) 9:6   
(d) 16:9
Solution. (d)
Ratio of volumes of two sphere is given as,

Exercise 12.2
Very Short Answer Type Questions:
Write whether True or False and justify your answer.
Question. 1 Two identical solid hemispheres of equal base radius r cm are stuck together along their bases. The total surface area of the combination is 6πr².
Solution. False
Explanation:
When two solid hemispheres of same base radius r are joined together along their bases then the new solid formed is a sphere with the same radius r.
∴Total surface area of sphere = 4πr²
Question. 2 A solid cylinder of radius r and height h is placed over other cylinder of same height and radius. The total surface area of the shape so formed is 4πrh + 4πr².
Solution. False
Explanation:
When a solid cylinder of radius r and height h is placed over other cylinder of same height and radius, we will egt a new cylinder with same radius r and height 2h.

Question. 3 A solid cone of radius r and height h is placed over a solid cylinder having same base radius and height as that of a cone. The total surface area of the combined solid is
Solution. False
Explanation:
Here a solid cone of radius r and height h is placed over a solid cylinder having same base radius and height as that of a cone.

Total surface area of the combined solid = Curved surface area of cone + Curved surface area of cylinder + Area of the base of cylinder      

Question. 4 A solid ball is exactly fitted inside the cubical box of side a. The volume of the ball is4/3πr³.
Solution. False
Explanation:


Solution. False
Explanation:
Given, a frustum of cone with h its vertical height and r₁ and r₂ are the radii of its two ends.




Solution. True
Explanation:

Solution. False
Explanation:
Given, a frustum withits vertical height h and r₁ and r₂ the radii of its two ends.

Question. 8 An open metallic bucket is in the shape of a frustum of a cone, mounted on a hollow cylindrical base made of the same metallic sheet. The surface area of the metallic sheet used is equal to curved surface area of frustum of a cone + area of circular base + curved surface area of cylinder.’
Solution. True
Explanation:
Given, a bucket which is in shape of a frustum is mounted on a hollow cylindrical base.

The surface area of the metallic sheet used to form given vessel will be equal to the total surface area of cylinder excluding the top and the curved surface area of frustrum of a cone.
∴ Total surface area of vessel = Curved surface area of frustrum + Curved surface area of cylinder + Area of base cylinder (Circular base of cylinder)
Exercise 12.3 (Q. No. 1-7)
Question. 1 Three metallic solid cubes whose edges are 3cm, 4cm and 5cm are melted and formed into a single cube. Find the edge of the cube so formed.
Solution.
Given, edges of three solid cubes are 3cm, 4cm and 5cm, respectively.
So, volumes of of three solid cubes are (3)³, (4)³ and (5)³ = 27cm³, 64cm³ and 125cm³, respectively.
Now, Since volume of resulting cube = Sum of volume of three cubes = (27 + 64+ 125) = 216cm³
Let R be the edge of the resulting cube
⟹           R³ =216
⟹           R = 6cm
Hence, the edge of new cube is 6 cm.
Question. 2 How many shots each having diameter 3cm can be made from a cuboidal Lead solid of dimensions 9cm × 11cm × 12cm?
Solution.

Question. 3 A bucket is in the form of a frustum of a cone and holds 28.490 L of water. The radii of the top and bottom are 28cm and 21cm, respectively. Find the height of the bucket.
Solution.

Question. 4 A cone of radius 8cm and height 12cm 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.
Solution.
Let ABC be the given cone with radius 8cm and height 12cm.


Question. 5 Two identical cubes each of volume 64 cm³ are joined together end to end. What is the surface area of the resulting cuboid?
Solution.

Question. 6   From a solid cube of side 7cm, a conical cavity of height 7cm and radius 3cm is hollowed out. Find the volume of the remaining solid.
Solution.

Question. 7 Two cones with same base radius 8cm and height 15cm are joined together along their bases. Find the surface area of the shape so formed.
Solution.
When two identical cones are joined base to base, the total surface area of new solid becomes equal to the sum of curved surface areas of both the cones.
Given, radius of each cone, r = 8cm
And height of each cone, h = 15 cm

Exercise 12.3
Short Answer Type Questions (Q. No. 8-14):
Question. 8 Two solid cones A and B are placed in a cylindrical tube as shown in the figure. The ratio of their capacities is 2:1. Find the height and capacities of cones. Also, find the volume of the remaining portion of the cylinder.

Given, the ratio of their capacities is 2:1.
So, let volume of Cone Ibe 2V and volume of Cone IIbe V.
Again, let height of the Cone I = h₁ cm
Then height of Cone II = (21 – h₁) cm
Given, diameter of the cones = 6cm


Question. 9 An ice-cream cone full of ice-cream having radius 5cm and height 10cm as shown in figure given below. Calculate the volume of ice-cream, provided that its 1/6 part is Left unfilled with ice-cream.

Solution.
Given, ice-cream cone is the combination of a cone of radius 5 cm and height 10 cm and a hemisphere of radus 5 cm.

Question. 10 Marbles of diameter1.4cm are dropped into a cylindrical beaker of diameter 7cm containing some Water. Find the number of marbles that should be dropped into the beaker, so that the water level rises by 5.6cm.
Solution.
When marbles are dropped in beaker containing some water, the volume of water raised in beaker will be equal to the volume of total marbles.

Question. 11 How many spherical lead shots each of diameter 4.2cm can be obtained from a solid rectangular lead piece with dimensions 66cm, 4cm and 21cm?
Solution.

Hence, the required number of spherical lead shots is 1500.
Question. 12 How many spherical lead shots of diameter 4cm can be made out of a solid cube of lead whose edge measures 44cm.
Solution.

Question. 13 A wall 24m long, 0.4m thick and 6m high is constructed with the bricks in each of the dimensions 25cm × 16cm × 10cm. If the mortar occupies 1/10th of the volume of the wall, then find the number of bricks used in constructing the wall.
Solution.
Given, length of wall, l = 24m,
Thickness of wall, b = 0.4m, and
Height of wall, h = 6m


Hence, the required number of bricks used in constructing the wall is 12960.
Question. 14 Find the number of metallic circular disc with 1.5cm base diameter and of height 0.2cm to be melted to form a right circular cylinder of height 10 cm and diameter 4.5cm.
Solution.
Given, height of each circular disc, i.e., a circular cylinder, h = 0.2cm
And diameter of each circular disc = 1.5cm

Exercise 12.4
Long Answer Type Questions (Q. No. 1-10)
Question. 1 A solid metallic hemisphere of radius 8 cm is melted and recasted into a right circular cone of Base radius 6cm. Determine the height of the cone.
Solution.

Hence, the required height of cone, h = 28.44cm
Question. 2 A rectangular water tank of base 11m × 6m contains water upto a height of 5m. If the water in the tank is transferred to a cylindrical tank of radius 3.5m, find the height of the water level in the tank.
Solution.
Given, height of water in a length, rectangular tank of base dimensions 11m and 6m is 5m.
Therefore, volume of the water in rectangular tank = 11 ×6 × 5= 330m³
Also, given radius of the cylindrical tank, r = 3.5m
Let height of water level in cylindrical tank be h.

Hence, the height of water level in cylindrical tank is 8.6m.
Question. 3 How many cubic centimetres of iron is required to construct an open box whose external dimensions are 36cm, 25cm and 16.5cm provided the thickness of the iron is 1.5cm. If one cubic centimetre of iron weights 7.5g, then find the weight of the box.
Solution.
Given, external dimensions of a box are 36cm × 25cm × 16.5cm.
Therefore, external volume of an open box = 36 × 25 × 16.5 = 14850cm³

Question. 4 The barrel of a fountain pen, cylindrical in shape, is 7cm long and 5mm in diameter. A full barrel of ink in the pin is used up on writing 3300 words on an average. How many words can be written in a bottle of ink containing one-fifth of a litre?
Solution.
Given, length of the barrel of a fountain pen = 7cm

Question. 5 Water flows at the rate of 10m min^-1 through a cylindrical pipe 5m in diameter. How long would it take to fill a conical vessel whose diameter at the base is 40cm and depth 24cm?
Solution.
Given, speed of water flow = 10m min^-1 = 1000 cm/min

Question. 6 A heap of rice is in the form of a cone of diameter 9m and height 3.5m. Find the volume of the rice. How much canvas cloth is required to just cover heap?
Solution.
Given, height of the conical heap of rice, h = 3.5m
And diameter of the conical heap of rice = 9m
∴ Radius of the conical heap of rice, r = 9/2 m

Hence, 80.61 m² canvas cloth is required to just cover the heap of rice.
Question. 7 A factory manufactures 120000 pencils daily. The pencils are cylindrical in shape each of length 25cm and circumference of base as 1.5cm. Determine the cost of colouring the curved surfaces of the pencils manufactured in one day at  Rs.0.05 per dm².
Solution.
Given, length of one pencil which is cylindrical in shape, h = 25cm
And circumference of base of pencil = 1.5 cm

Question. 8 Water is flowing at the rate of 15kmh⁻¹ through a pipe of diameter 14cm into a cuboidal pond which is 50m long and 44m wide. In what time will the level of water in pond rise by 21cm?
Solution.
Given, speed of water flowing through the pipe, h = 15kmh⁻¹ = (15 × 1000) = 15000 mh^-1
Length of the pond, l= 50m
And, width of the pond, b = 44m
And, level upto which water is to be filled, h = 21 cm = 21/100 m

Question. 9 A solid iron cuboidal block of dimensions 4.4m × 2.6m × 1m is recast into a hollow cylindrical pipe of internal radius 30cm and thickness 5cm. Find the length of the pipe.
Solution.
Here, volume of a solid iron cuboidal block = 4.4m × 2.6m × 1m = 11.44m³
Also, internal radius of hollow cylindrical pipe, r₁ = 30cm = 0.3m
And thickness of hollow cylindrical pipe = 5cm = 0.05m
So, external radius of hollow cylindrical pipe, r₂ = r₁+ Thickness = 0.3 + 0.05 = 0.35m
Let length of hollow cylindrical pipe be h m.

Question. 10 500 persons are taking a dip into a cuboidal pond which is 80m Long and 50m broad. What is the rise
of water level in the pond, if the average displacement of the water by a person is 0.04m³ ?
Solution.
Let the rise of water level in the pond be hm, when 500 persons are taking a dip into a cuboidal pond.
Given, length of the cuboidal pond = 80m
And breadth of the cuboidal pond = 50m
Now, volume for the rise of water level in the pond = Length × Breadth ´ Height
                        = 80 × 50 ´ h = 4000hm³
Given that average displacement of the water by a person = 0.04m³
So, the average displacement of the water by 500 persons = 500 ´ 0.04m³
So, according to the given condition,
Volume for the rise of water level in the pond = Average displacement of the water by 500 persons

Hence, the required rise of water level in the pond is 0.5cm.
Exercise 12.4
Long Answer Type Questions (Q. No. 11-14):
Question. 11 16 glass spheres each of radius 2cm are packed into a cuboidal box of internal dimensions 16cm × 8cm × 8cm and then the box is filled with water. Find the volume of water filled in the box.
Solution.

Question. 12 A milk container of height 16cm is made of metal sheet in the form of a frustum of a cone with radii of its lower and upper ends as 8cm and 20cm, respectively. Find the cost of milk at the rate of ` 22 per L
which the container can hold.
Solution.              
Given, radius of lower end of milk container, r = 8 cm
Radius of upper end of milk container, R = 20cm
And, height of milk container, h = 16cm
Now, as the container is in the shape of a a frustum of a cone,

Now, given that cost of 1 Lmilk = Rs.22
⟹ Cost of 10.45942 L milk = 22 × 1045942 = Rs.230.12
Hence, the required cost of milk contained in the container is Rs.230.12.
Question. 13 A cylindrical bucket of height 32cm and base radius 18cm is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is 24cm, find the radius and slant height of the heap.
Solution.
Given, radius of the base of the bucket = 18cm
And, height of the bucket = 32cm


Question. 14 A rocket is in the form of a right circular cylinder closed at the lower end and surmounted by a
cone with the same radius as that of the cylinder. The diameter and height of the cylinder are 6cm and 12cm,
respectively. If the slant height of the conical portion is 5cm, then find the total surface area and volume of the
rocket, (use p= 3.14)
Solution.
Since, rocket is the combination of a right circular cylinder and a cone.


And total surface area of the rocket = CSA of cone + CSA of cylinder + Area of base of cylinder
                                                    = 47.1 + 226.08+28.26 = 301.44cm²