RD SHARMA SOLUTION CHAPTER –21 Mensuration – II (Volumes and Surface Areas of a| CLASS 8TH MATHEMATICS-EDUGROWN

Exercise 21.1

Question 1.
Find the volume of a cuboid whose
(i) length = 12 cm, breadth = 8 cm, height = 6 cm
(ii) length = 1.2 m, breadth = 30 cm, height = 15 cm
(iii) length = 15 cm, breadth = 2.5 dm, height = 8 cm

Solution:
In a cuboid,
(i) Length (l) = 12 cm
Breadth (b) = 8 cm
Height (h) = 6 cm
∴ Volume = Ibh = 12 x 8 x 6 cm³ = 576 cm³
(ii) Length (l) = 1.2 m = 120 cm
breadth (6) = 30 cm
Height (h) = 15 cm
∴ Volume = Ibh = 120 x 30 x 15 cm³ = 54000 cm³
(iii) Length (l) = 15 cm
Breadth (b) = 2.5 dm = 25 cm
Height (h) = 8 cm
∴ Volume = Ibh
= 15 x 25 x 8 cm³ = 3000 cm²

Question 2.
Find the volume of the cube whose side is
(i) 4 cm
(ii) 8 cm
(iii) 1.5 dm
(iv) 1.2 m
(v) 25 mm.
Solution:
(i) Side of a cube (a) = 4 cm
∴ Volume = a3 = (4)³ cm³ = 4 x 4 x 4 = 64 cm³

(ii) Side of cube (a) = 8 cm
∴ Volume = a3 = (8)³ 4 cm
= 8 x 8 x 8 cm³ = 512 cm³
(iii) Side of cube (a) = 1.5 dm = 15 cm
∴ Volume = a³ = (1.5)3 dm² = (15)³ cm³
= 15 x 15 x 15 = 3375 cm³
(iv) Side of cube (a) = 1.2 m = 120 cm
∴ Volume = a³ = (120)³ cm³
= 120 x 120 x 120 = 1728000 cm³
(v) Side of cube (a) = 25 mm = 2.5 cm.
∴ Volume = a³ = (2.5)³ cm³
= 2.5 x 2.5 x 2.5 cm³ = 15.625 cm³

Question 3.
Find the height of a cuboid of volume 100 cm3 whose length and breadth are 5 cm and 4 cm respectively.
Solution:
Volume of a cuboid =100 cm³
Length (1) = 5 cm
and breadth (b) = 4 cm

Question 4.
A cuboidal vessel is 10 cm long and 5 cm wide, how high it must be made to hold 300 cm3 of a liquid ?
Solution:
Volume of the liquid in the vessel = 300 cm³
Length (l)= 10 cm
Breadth (b) = 5 cm

Question 5.
A milk container is 8 cm long and 50 cm wide. What should be its height so that it can hold 4 litres of milk ?
Solution:
Capacity of milk = 4 litres
∴ Volume of the container = 4 x 1000 cm³ = 4000 cm³
Length (l) = 8 cm
Width (b) = 50 cm

Question 6.
A cuboidal wooden block contains 36 cm³ wood. If it be 4 cm long and 3 cm wide, find its height.
Solution:
Volume of wooden cuboid block = 36 cm³
Length (l) = 4 cm
Breadth (b) = 3 cm

Question 7.
What will happen to the volume of a cube, if its edge is (i) halved (ii) trebled ?
Solution:
Let side of original cube = a cm
∴ Volume = a³ cm³
(i) In first case,

(ii) In second case, when side (edge) is trebled, then side = 3a
∴ Volume = (3a)³ = 27a³
∴ It will be 27 times

Question 8.
What will happen to the volume of a cuboid if its (i) Length is doubled, height is same and breadth is halved ? (ii) Length is doubled, height is doubled and breadth is same ?
Solution:
Let l, b and h be the length, breadth and height of the given cuboid respectively.
∴ Volume = lbh.
(i) Length is doubled = 21

∴ The volume will be the same.
(ii) Length is doubled = 21
breadth is same = b height is doubled = 2h
∴ Volume = 2l x b x 2h = 4 lbh
∴ Volume will be 4 times

Question 9.
Three cuboids of dimensions 5 cm x 6 cm x 7 cm, 4 cm x 7 cm * 8 cm and 2 cm x 3 cm x 13 cm are melted and a cube is made. Find the side of cube.
Solution:
Dimensions of first cuboid = 5 cm x 6 cm x 7 cm
∴ Volume = 5 x 6 x 7 = 210 cm³
Dimensions of second cuboid = 4 cm x 7 cm x 8 cm
∴ Volume = 4x 7 x 8 = 224 cm³
Dimensions of third cuboid = 2 cm x 3 cm x 13 cm
∴ Volume = 2 x 3 x 13 = 78 cm³
Total volume of three cubes = 210 + 224 + 78 cm3 = 512 cm³
∴ Volume of cube = 512 cm³

Question 10.
Find the weight of solid rectangular iron piece of size 50 cm x 40 cm x 10 cm, if 1 cm3 of iron weighs 8 gm.
Solution:
Dimension of cuboidal iron piece = 50 cm x 40 cm x 10 cm
∴ Volume = 50 x 40 x 10 = 20000 cm³
Weight of 1 cm³ = 8 gm
∴ Total weight of piece = 20000 x 8 gm

Question 11.
How many wooden cubical blocks of side 25 cm can be cut from a log of wood of size 3 m by 75 cm by 50 cm, assuming that there is no wastage ?
Solution:
Length of log (l) = 3 m = 300 cm.
Breadth (b) = 75 cm
and height (h) = 50 cm
∴ Volume of log = lbh = 300 x 75 x 50 cm3 = 1125000 cm³
Side of cubical block = 25 cm
∴ Volume of one block = a² = 25 x 25 x 25 cm3 = 15625 cm³
∴ Number of blocks to be cut out

Question 12.
A cuboidal block of silver is 9 cm long, 4 cm broad and 3.5 cm in height. From it, beads of volume 1.5 cm² each are to be made. Find the number of beads that can be made from the block ?
Solution:
Length of block (l) = 9 cm
Breadth (b) = 4 cm
and height (h) = 3.5 cm
∴ Volume = l x b x h = 9 x 4 x 3.5 cm³ = 126 cm³
Volume of one bead = 1.5 cm³
∴ Number of beads = 126105 = 84

Question 13.
Find the number of cuboidal boxes measuring 2 cm by 3 cm by 10 cm which can be stored in a carton whose dimensions are 40 cm, 36 cm and 24 cm.
Solution:
Length of cuboidal box (l) = 2 cm
breadth (b) = 3 cm
and height (h) = 10 cm
∴ Volume = lx b x h = 2 x 3 x 10 = 60 cm³
Volume of carton = 40 x 36 x 24 cm³
= 34560 cm³
∴ Number of boxes to be height in the carton

Question 14.
A cuboidal block of solid iron has dimensions 50 cm, 45 cm and 34 cm. How many cuboids of size 5 cm by 3 cm by 2 cm can be obtained from the block ? Assume cutting causes no wastage.
Solution:
Dimensions of block = 50 cm, 45 cm, 34 cm
∴ Volume = 50 x 45 x 34 = 76500 cm³
Size of cuboid = 5 cm x 3 cm x 2 cm
∴ Volume of cuboid =  5 x 3 x 2 = 30 cm³

Question 15.
A cube A has side thrice as long as that of cube B ? What is the ratio of the volume of cube A to that of cube B ?
Solution:
Let side of cube B = a
Then Volume = a³
and side of cube A = 3a
Volume = (3a)³ = 3a x 3a x 2a = 27a³
∴ Ratio of volume’s A and B = 27a³ : a³
= 27 : 1

Question 16.
An ice-cream brick measures 20 cm by 10 cm by 7 cm. How many such bricks can be stored in a deep fridge whose inner dimensions are 100 cm by 50 cm by 42 cm ?
Solution:
Dimensions of ice cream brick = 20 cm x 10 cm x 7 cm
∴ Volume = 20 x 10 x 7 cm³ = 1400 cm³
Dimensions of inner of fridge = 100 cm x 50 cm x 42 cm = 210000 cm³
∴ Number of bricks to be kept in the fridge

Question 17.
Suppose that there are two cubes, having edges 2 cm and 4 cm, respectively. Find the volume V₁ and V₂ of the cubes and compare them.
Solution:
Side of first cube (a) = 2 cm
∴ Volume (V₁) = a³ = (2) = 8 cm³
Similarly side of second cube = 4 cm
and volume (V₂) = (4)³ = 64 cm³
Now V₂ = 64 cm³ = 8 x 8 cm³
= 8 x V₁
⇒ V₂ = 8V₁

Question 18.
A tea-packet measures 10 cm x 6 cm x 4 cm.How many such tea-packets can be placed in a cardboard box of dimensions 50 cm x 30 cm x 0.2 m ?
Solution:
Dimensions of tea-packet = 10cm x 6cm x 4 cm
∴ Volume =10 x 6 x 4 = 240 cm³
Dimensions of box = 50 cm x 30 cm x 0.2 m
= 50 cm x30 cm x20 cm
∴ Volume = 50 x 30 x 20 = 30000 cm³
∴ Number of tea-packets to be kept = 30000240

Question 19.
The weight of a metal block of size 5 cm by 4 cm by 3 cm is 1 kg. Find the weight of a block of the same metal of size 15 cm by 8 cm by 3 cm.
Solution:
Dimensions of a metal block = 5 cm x 4 cm x 3 cm = 5 x 4 x 3 = 60 cm³
Dimensions of a second block = 15 cm x 8 cm x 3 cm = 15 x 8 x 3 = 360 cm³
But weight of first block = 1 kg
∴ Weight of second block
= 116 x 360 = 6 kg

Question 20.
How many soap cakes can be placed in a box of size 56 cm x 0.4 m x 0.25 m, it the size of soap cake is 7 cm x 5 cm x 2.5 cm ?
Solution:
Size of box = 56 cm x 0.4 m x 0.25 m = 56 cm x 40 cm x 25 cm
∴ Volume = 56 x 40 x 25 cm³ = 56000 cm³
Size of a soap cake = 7 cm x 5 cm x 2.5 cm
∴ Volume = 7 x 5 x 2.5 cm³ = 87.5 cm³
∴ Number of cakes to be kept in the box
= 5600087.5 = 640

Question 21.
The volume of a cuboid box is 48 cm3. If its height and length are 3 cm and 4 cm respectively, find its breadth.
Solution:
Volume of cuboid box = 48 cm³
Length (l) = 4 cm
Height = (h) = 3 cm

Exercise 21.2

Question 1.
Find the volume in cubic metres (cu.m) of each of the cuboids whose dimensions are :
(i) length = 12 cm, breadth = 10 m, height = 4.5 m
(ii) length = 4 m, breadth = 2.5 m, height = 50 cm
(iii) length = 10 m, breadth = 25 dm, height = 25 cm.
Solution:
(i) Length of cuboid (l) = 12 m
Breadth (b) = 10m
and height (h) = 4.5 m
∴Volume = l x b x h = 12 x 10 x 4.5 m³
= 540 m³
(ii) Length of cuboid (l) = 4 m
Breadth (b) = 2.5m
Height (h) = 50 cm = 0.5 m
∴ Volume = l x b x h = 4 x 2.5 x 0.5 = 5 m³
(iii) Length of cuboid (l) = 10 m
Breadth (b) = 25 dm = 2.5 m
Height (h) = 25 cm 0.25 m
∴ Volume = l x b x h = 10 x 2.5 x 0.25 m³ = 6.25 m³

Question 2.
Find the volume in cubic decimetre of each of the cubes whose side is
(i) 1.5 m
(it) 75 cm
(iii) 2 dm 5 cm
Solution:
(i) Side of cube (a) = 1.5 m
∴ Volume = a³ = (1.5)³ m³
= 1.5 x 1.5 x 1.5 m³ = 3.375 m³
= 3.375 x 1000 = 3375 dm³
(ii) Side of cube (a) = 75 cm = 7.5 dm
∴ Volume = a³ = (7.5)3 dm³
= 421.875 dm³
(iii) Side of cube (a) = 2 dm 5 cm = 2.5 dm
∴ Volume = (a)³ = (2.5)³ dm³
= 15.625 dm³

Question 3.
How much clay is dug out in digging a well measuring 3m by 2m by 5m?
Solution:
Length of well (l) = 3m
breadth (b) = 2 m
and height (depth) (h) = 5 m
Volume of earth dug out = l x b x h = 3 x 2 x 5 = 30m³

Question 4.
What will be the height of a cuboid of volume 168 m³, if the area of its base is 28 m² ?
Solution:
Volume of a cuboid = 168 m³
Area of its base l.e., l x b = 28 m³

Question 5.
A tank is 8 m long, 6 m broad and 2 m high. How much water can it contain ?
Solution:
Length of tank (l) = 8 m
Breadth (b) = 6 m
Height (h) = 2 m
∴ Volume of water in the tank = l x b x h = 8 x 6 x 2 = 96 m³
= 96 x 1000 = 96000litres (∵1m³ = 1000litre)

Question 6.
The capacity of a certain cuboidal tank is 50000 litres of water. Find the breadth of the tank if its height and length are 10 m and 2.5 m respectively.
Solution:
Capacity of water in the tank = 50000 litres
∴ Volume of water = 50000 x 11000 = 50 m³ (1000 l = 1 m³)
Height of tank (h)= 10 m
and length (l) = 2.5 m
Volume 50

Question 7.
A rectangular diesel tanker is 2 m long, 2 m wide and 40 cm deep. How many litres of diesel can it hold ?
Solution:
Length of tanker (l) = 2 m
Breadth (b) = 2m
Depth (h) = 40 cm = 0.4 m
∴ Volume = l x bx h = 2 x 2 x 0.4=1.6m³
Quantity of diesel = 1.6 x 1000 litres (1 m³= 1000 l)
= 1600 litres

Question 8.
The length, breadth and height of a room are 5 m, 4.5 m and 3 m, respectively. Find the volume of the air it contains.
Solution:
Length of room (l) = 5 m
Breadth (6) = 4.5 m
and height (h) = 3 m
∴ Volume of air it contains
= l x b x h = 5 x 4.5 x 3 m³
= 67.5 m³

Question 9.
A water tank is 3 m long, 2 m broad and 1 m deep. How many litres of water can it hold ?
Solution:
Length of tank (l) = 3 m
Breadth (b) = 2 m
and depth (h) = 1 m
∴ Volume of tank = l x b x h
= 3 x 2 x 1 = 6 m³
∴ Quantity of water it can contains
= 6 x 1000 litres = 6000 litres (1 m³= 1000 litres)

Question 10.
How many planks each of which is 3 m long, 15 cm broad and 5 cm thick can be prepared from a wooden block 6 m long, 75 cm broad and 45 cm thick ?
Solution:
Length of wooden block (l) = 6 m
Width (b) = 75 cm = 0.75 m
Thickness (h) = 45 cm = 0.45 m
∴ Volume = l x b x h = 6 x 0.75 x 0.45 m³
Length of plank (l) = 3 m
Breadth (b) = 15 cm = 0.15 m
Thickness (h) = 5 cm = 0.05 m
∴ Volume = 3 x 0.15 x 0.05 m³
Number of planks

Question 11.
How many bricks each of size 25 cm x 10 cm x 8 cm will be required to build a wall 5 m long, 3 m high and 16 cm thick assuming that the volume of sand and cement used in the construction is negligible ?
Solution:
Size of one brick = 25 cm x 10 cm x 8 cm
∴ Volume of one brick = 25 x 10 x 8 cm³

Length of wall (l) = 5 m
Width (b) = 0.16 m
Height (h) = 3 m
∴ Volume of wall = l x b x h
= 5 x 0.16 x 3 m³ = 2.4 m³
∴ Number of bricks required

Question 12.
A village, having a population of 4000 requires 150 litres water per head per day. It has a tank which is 20 m long, 15 m broad and 6 m high. For how many days the water of this tank will last ?
Solution:
Total population of a village = 4000
Water required for each person for one day = 150 litres
∴ Water required for 4000 persons for one day = 150 x 4000 = 600000 litres
Length of tank (l) = 20 m
Breadth (b) = 15 m
Height (h) = 6 m
∴ Volume of tank = l x b x h = 20 x 15 x 6 m³ = 1800 m³
Capacity of water in the tank = 1800 x 1000 l= 1800000l (1 m³ = 1000 l)
∴ Number of days, the water will last

Question 13.
A rectangular field is 70 m long and 60 m broad. A well of dimensions 14 m x 8 m x 6 m is dug outside the field and the earth dugout from this well is spread evenly on the field. How much will the earth level rise ?
Solution:
Length of well (l) = 14 m
Breadth (A) = 8m
Depth (A) = 6m
∴ Volume of earth dugout = l x bx h
= 14 x 8 x 6 = 672 m³ Length of field = 70 m
and breadth = 60 m
Let h be the height of earth spread over
Then 70 x 60 x h = 672
⇒ h = 67270×60 = 0.16m
∴ Height of earth = 0.16 m = 16 cm

Question 14.
A swimming pool is 250 m long and 130 m wide. 3250 cubic metres of water is pumped into it. Find the rise in the level of water.
Solution:
Volume of water = 3250 m³
Length of pool (l) = 250 m
Breadth (b)= 130 m
∴ Height of water level

Question 15.
A beam 5 m long and 40 cm wide contains 0.6 cubic metres of wood. How thick is the beam?
Solution:
Volume of wood of the beam = 0.6 m³ = 600000
Length of beam (l) = 5 m = 500 cm
Breadth (b) = 40 cm

Question 16.
The rainfall on a certain day was 6 cm. How many litres of water fell on 3 hectares of field on that day ?
Solution:
Area of the field = 3 hectares
= 3 x 10000 square metres
= 30000 square metres
Height of rainfall = 6 cm = m³

Question 17.
An 8 m long cuboidal beam of wood when sliced produces four thousand 1 cm cubes and there is no wastage of wood in this process. If one edge of the beam is 0.5 m, find the third edge.
Solution:
Length of cuboidal beam (l) = 8 m = 800 cm
Number of cubical sliced = 4000
Edge of each cube = 1 cm
Volume of beam = 4000 (1)³ cm³ = 4000 cm³
One edge of the beam = 0.5 m = 50 cm.

Question 18.
The dimensions of a metal block are 2.25 m by 1.5 m by 27 cm. It is melted and recast into cubes, each of the side 45 cm. How many cubes are formed ?
Solution:
Dimensions of metal block = 2.25 m x 1.5 m x 27 cm
∴ Volume = 2.25 x 1.5 x 0.27 m³
= 225 x 150 x 27 cm³ = 911250 cm³
Side of each cube (a) = 45 cm
∴ Volume of one cube = a³ = (45)³ cm³ = 91125 cm³
∴ Number of cubes = 91125091125 = 10

Question 19.
A solid rectangular piece of iron measures 6 m by 6 cm by 2 cm. Find the weight of this piece if 1 cm3 of iron weighs 8 gm.
Solution:
Dimensions of a piece of rectangular iron = 6m x 6cm x 2cm
∴ Volume = 600 x 6 x 2 cm³ = 7200 cm³
Weight of 1 cm³ = 8 gm
∴ Total weight of the piece = 7200 x 8 gm
= 57600 gm = 576001000 kg = 57.6 kg

Question 20.
Fill in the blanks in each of the following so as to make the statement true :
(i) 1 m³ = ……… cm³
(ii) 1 litre = …….. cubic decimetre
(iii) 1 kl = …… m³
(iv) The volume of a cube of side 8 cm is …….. .
(v) The volume of wooden cuboid of length 10 cm and breadth 8 cm is 4000 cm3. The height of the cuboid is …….. cm
(vi) 1 cu.dm = …….. cu.mm
(vii) 1 cu.km = ……cu.m
(viii) 1 litre =…….. cu.cm
(ix) 1 ml = ……… cu.cm
(x) 1 kl = ……… cu.dm = ……. cu.cm.
Solution:
(i) 1 m³ = 1000000 or 10⁶ cm³
(ii) 1 litre = 1 cubic decimetre
(iii) 1 kl = 1 m³
(iv) The volume of a cube of side 8 cm is 512 cm³ (V = a³ = 8 x 8 x 8 = 512 cm³)
(v) The volume of a wooden cuboid of length 10 cm and breadth 8 cm is 4000 cm³. The height of the cuboid is 50 cm

(vi) 1 cu.dm = 1000000 cu mm = 10⁶ cu.mm
(vii) 1 cu.km = 1000 x 1000 x 1000 cu.m = 109 cu.m
(viii) 1 litre = 1000 cu.cm = 10³ cu.cm
(ix) 1 ml = 1 cu.cm
(x) 1 kl = 1000 cu.dm = 100 x 100 x 100 cu.cm = 10⁶ cu.cm

Exercise 21.3

Question 1.
Find the surface area of a cuboid whose :
(i) length = 10 cm, breadth = 12 cm and height = 14 cm
(ii) length = 6 dm, breadth = 8 dm, height = 10 dm
(iii) length = 2 m, breadth = 4 m and height = 5 m
(iv) length = 3.2 m, breadth = 30 dm, height = 250 cm.
Solution:
(i) Length of cuboid (l) = 10 cm
Breadth (b) = 12 cm
Height (h) = 14 cm
∴ Surface area = 2(1 × b + b × h + h × l)
= 2(10 x 12 + 12 x 14 + 14 x 10) cm²
= 2(120+ 168 + 140) cm²
= 2 x 428 = 856 cm²
(ii) Length of cuboid (l) = 6 dm
Breadth (b) = 8 dm
Height (h) = 10 dm
∴ Surface area = 2 ( l × b + b x h + h× l)
= 2(6 x 8 + 8 x 10 + 10 x 6) dm²
= 2(48 + 80 + 60) dm² = 2 x 188 = 376 dm²
(iii) Length of cuboid (l) = 2 m
Breadth (b) = 4 m
Height (h) = 5 m
∴ Surface area = 2(l × b + b × h + h × l)
= 2(2 x 4 + 4 x 5 + 5 x 2) m2
= 2(8 + 20 + 10) m2 = 76 m2
(iv) Length of cuboid (l) = 3.2 m = 32 dm
Breadth (b) = 30 dm
Height (h) = 250 cm = 25 dm
∴ Surface area = 2(1 x b + b x h + h x l)
= 2(32 x 30 + 30 x 25 + 25 x 32) dm²
= 2(960 + 750 + 800) dm²
= 2 x 2510 = 5020 dm²

Question 2.
Find the surface area of a cube whose edge is
(i) 1.2 m
(ii) 27 cm
(iii) 3 cm
(iv) 6 m
(v) 2.1m
Solution:
(i) Edge of the cube (a) = 1.2 m
∴ Surface area = 6a²= 6 x (1,2)² m²
= 6 x 1.44 = 8.64 m²
(ii) Edge of cube (a) = 27 cm
∴ Surface area = 6a² = 6 x (27)² m²
= 6 x 729 = 4374 m²
(iii) Edge of cube (a) = 3 cm
Surface area = 6a² = 6 x (3)² m²
= 6×9 cm² = 54 cm²
(iv) Edge of cube (a) = 6 m
∴ Surface area = 6a² = 6 x (6)2 m²
= 6 x 6 x 6 = 216 m²
(v) Edge of the cube (a) = 2.1 m
∴ Surface area = 6a2 = 6 x (2.1)2 m²
= 6 x 4.41 = 26.46 m²

Question 3.
A cuboidal box is 5 cm by 5 cm by 4 cm. Find its surface area.
Solution:
Length of cuboid box (l) = 5 cm
Breadth (b) = 5 cm
and height (h) = 4 cm
∴ Surface area = 2 (l x b + b x h + h x l)
= 2 (5 x 5 + 5 x 4 + 4 x 5) cm²
= 2 (25 + 20 + 20)
= 2 x 65 cm²
= 130 cm²

Question 4.
Find the surface area of a cube whose volume is :
(i) 343 m³
(ii) 216 dm³.
Solution:
(i) Volume of a cube = 343 m³

Question 5.
Find the volume of a cube whose surface area is
(i) 96 cm²
(ii) 150 m².
Solution:
(i) Surface area of a cube = 96 cm²

Question 6.
The dimensions of a cuboid are in the ratio 5:3:1 and its total surface area is 414 m². Find the dimensions.
Solution:
Ratio in .dimensions = 5 : 3 : 1
Let length (l) = 5x
breadth (b) = 3x
and height (h) = x
∴ Surface area = 2(1 x b + b x h + h x l)
= 2(5x x 3x + 3x x x + x x 5x)
= 2(15×2 + 3×2 + 5×2) = 2 x 23×2 = 46×2

Question 7.
Find the area of the cardboard required to make a closed box of length 25 cm, 0.5 m and height 15 cm.
Solution:
Length of cardboard (l) = 25 cm
Breadth (b) = 0.5 m = 50 cm
Height (h)= 15 cm.
∴ Surface area of cardboard = 2 (l x b + b x h + h x l)
= 2(25 x 50 + 50 x 15 + 15 x 25) cm²
= 2(1250+ 750+ 375) cm²
= 2(2375)
= 4750 cm²

Question 8.
Find the surface area of a wooden box whose shape is of a cube and if the edge of the box is 12 cm.
Solution:
Edge of cubic wooden box = 12 cm
∴ Surface area = 6a² = 6(12)² cm²
= 6 x 144 = 864 cm²

Question 9.
The dimensions of an oil tin are 26 cm x 26 cm x 45 cm. Find the area of the tin sheet required for making 20 such tins. If 1 square metre of the tin sheet costs Rs 10, find the cost of the tin sheet used for these 20 tins.
Solution:
Length of tin (l) = 26 cm = 0.26 m
Breadth (b) = 26 cm = 0.26 m
Height (h) = 45 cm = 0.45 m
∴ Surface area = 2(l x b + b x h +h xl)
= 2(0.26 x 0.26 + 0.26 x 0.45 + 0.45 x 0.26) m²
= 2(0.0676 + 0.117 + 0.117) m²
= 2(0.3016) = 0.6032 m²
Sheet required for such 20 tins
= 0.6032 x 20= 12.064 m²
Cost of 1 m² tin sheet = 10 m
∴ Total cost = Rs 12.064 x 10 = Rs 120.64
and area of sheet = 12.064 m² = 120640 cm²

Question 10.
A classroom is 11 m long, 8 m wide and 5 m high. Find the sum of the areas of its floor and the four walls (including doors, windows etc.)
Solution:
Length of room (l) = 11 m
Width (b) = 8 m
and height (h) = 5 m
Area of floor = l x b = 11 x8 = 88m2
Area of four walls = 2 (l + b) x h
= 2(11 + 8) x 5 m² = 2 x 19×5 = 190 m²
∴ Total area = 88 m² + 190 m² = 278 m²

Question 11.
A swimming pool is 20 m long, 15 m wide and 3 m deep. Find the cost of repairing the floor and wall at the rate of Rs 25 per square metre.
Solution:
Length of pool (l) = 20 m
Breadth (b) = 15 m
and Depth (h) = 3 m.
Area of floor = l x b = 20 x 15 = 300 m²
and area of its walls = 2(l + b) x h
= 2(20 + 15) x 3 = 2 x 35 x 3 m² = 210 m²
∴ Total area = 300 + 210 = 510 m²
Rate of repairing it = Rs 25 per sq. me

Question 12.
The perimeter of a floor of a room is 30 m and its height is 3 m. Find the area of four walls of the room.
Solution:
Perimeter of floor = 30 m
i.e. 2(1 + b) = 30 m
Height = 3 m
∴ Area of four walls = Perimeter x height = 30 x 3 = 90 m²

Question 13.
Show that the product of the areas of the floor and two adjacent walls of a cuboid is the square of its volume.
Solution:
Let length of the room = l
and breadth = b
and height = h
Volume = l x b x h
Area of floor = l x b = lb.
Area of two adjacent walls = hl x bh.
∴ Product of areas of floor and two adjacent walls of the room = lb (hi x bh)
= l²b²h² = (l.b.h)² = (Volume)²
Hence proved

Question 14.
The walls and ceiling of a room are to be plastered. The length, breadth and height of the room are 4.5, 3m and 350 cm, respectively. Find the cost of plastering at the rate of Rs 8 per square metre.
Solution:
Length of room (l) = 4.5 m
Width (b) = 3 m
and height (h) = 350 cm = 3.5 m
∴ Area of walls = 2(l + b) x h
= 2(4.5 + 3) x 3.5 m² = 2 x 7.5 x 3.5 m² = 52.5 m²
Area of ceiling = l x b = 4.5 x 3 = 13.5 m²
∴ Total area = 52.5 + 13.5 m² = 66 m²
Rate of plastering = Rs 8 per sq. m
∴ Total cost = Rs 8 x 66 = Rs 528

Question 15.
A cuboid has total surface area of 50 m² and lateral surface area its 30 m². Find the area of its base.
Solution:
Total surface area of cuboid = 50 m²
Lateral surface area = 30 m²
∴ Area of floor and ceiling = 50 – 30 = 20 m²
But area of floor = area of ceiling
∴ Area of base (floor) = 202 = 10 m²

Question 16.
A classroom is 7 m long, 6 m broad and 3.5 m high. Doors and windows occupy an area of 17 m². What is the cost of white-washing the walls at the rate of Rs 1.50 per m².
Solution:
Length of room (l) = 7 m
Breadth (b) = 6 m
and height (h) = 3.5 m
∴ Area of four walls = 2(1 + b) x h
= 2(7 + 6) x 3.5 m² = 2 x 13 x 3.5 = 91 m²
Area of doors and windows = 17 m2
∴ Remaining area of walls = 91 – 17 = 74 m²
Rate of whitewashing = Rs 1.50 per m²
∴ Total cost = 74 x Rs 1.50 = Rs 111

Question 17.
The central hall of a school is 80 m long and 8 m high. It has 10 doors each of size 3 m x 1.5 m and 10 windows each of size 1.5 m x l m. If the cost of the white-washing the walls of the hall at the rate of Rs 1.20 per m² is Rs 2385.60, find the breadth of the hall.
Solution:
Length of hall (l) = 80 m
Height (h) = 8 m
Size of each door = 3 m x 1.5 m
∴ Area of 10 doors = 3 x 1,5 x 10 m²
= 45 m²
A size of each windows = 1.5 m x 1 m
∴ Area of 10 windows = 1.5 m x 1 x 10= 15 m²
Total cost of whitewashing the walls = Rs 2385.60
Rate of whitewashing = Rs 1.20 per m²
∴ Area of walls which are whitewashed

Exercise 21.4

Question 1.
Find the length of the longest rod that can be placed in a room 12 m long, 9 m broad and 8 m high.
Solution:
Length of room (l) = 12m
Breadth (b) = 9 m
Height (h) = 8 m

Longest rod to be kept in the room

Question 2.
If V is the volume of the cuboid of dimensions a, b, c and S its the surface area then prove that

Solution:
∵ a, b, c are the dimensions of a cuboid
∴ Volume (V) = abc
Surface area (S) = 2(ab + bc + ca)
Now

Question 3.
The areas of three adjacent faces of a cuboid are .v, y and z. If the volume is V¹ prove that V² = xyz.
Solution:
Let length of cuboid = l
Breadth = b
and height = h
Volume = Ibh
∴ x = lb,y = bh and z = hl
Now x.y.z = lb.bh.hl
= l² b² h² = (Ibh)² = V²
∴ V² = xyz Hence proved

Question 4.
A rectangular water reservoir contains 105 m3 of water. Find the depth of the water in the reservoir if its base measures 12 m by 3.5 m.
Solution:
Volume of the water in reservoir = 105 m²
Length (l)= 12 m
and breadth (b) = 3.5 m

Question 5.
Cubes A, B, C having edges 18 cm, 24 cm and 30 cm respectively are melted and moulded into a new cube D. Find the edge of the bigger cube D.
Solution:
Edge of cube A = 18 cm
∴ Volume = a² = (18)3 cm³ = 5832 cm³
Edge of cube B = 24 cm
∴ Volume = (24)³ = 13824 cm³
Edge of cube C = 30 cm
∴Volume = (30)³ = 27000 cm³
Volume of A, B, C cubes
= 5832+ 138-24+ 27000 = 46656 cm³
Volume of cube D = 46656 cm³

Question 6.
The breadth of a room is twice its height, one half of its length and the volume of the room is 512 cu.dm. Find its dimensions.
Solution:
Volume of room = 512 cu.dm
Let height of the room (h) = x
Then breadth (b) = 2x
and length (l) = 2x x 2 = 4x.
∴ Volume = l x b x h = 4x x 2x x x = 8×3

Question 7.
A closed iron tank 12 m long, 9 m wide and 4 m deep is to be made. Determine the cost of iron sheet used at the rate of Rs 5 per metre sheet, sheet being 2 m wide.
Solution:
Length of iron tank (l) = 12 m
Breadth (b) = 9 m
Depth (h) = 4 m
∴ Surface area of the tank = 2(l x b + b x h + h x l)
= 2(12 x 9 + 9 x 4 + 4 x 12) m²
= 2(108 + 36 + 48) = 2 x 192 m²
= 384 m²
Width of sheet used = 2 m

Question 8.
A tank open at the top is made of iron sheet 4 m wide. If the dimensions of the tank are 12 m x 8 m x 6 m, find the cost of iron sheet at Rs 17.50 per metre.
Solution:
Dimensions of the open iron tank = 12mx 8m.x 6m
∴ Surface area (without top)
= 2(1 x b) x h + lb
= 2(12 + 8) x 6+12 x 8m²
= 2 x 20 x 6 + 96 = 240 + 96 m² = 336 m²
Width of sheet used = 4 m
∴ Length of sheet = Areab = 3364 m = 84 m b 4
Rate of sheet = Rs 17.50 per m.
∴ Total cost = Rs 17.50 x 84 = Rs 1470

Question 9.
Three equal cubes are placed adjacently in a row. Find the ratio of the total surface area of the new cuboid to that of the sum of the surface areas of the three cubes.
Solution:
Let edge of each equal cubes = x
Then, surface area of one cube = 6x²

and surface area of three cubes = 3 x 6x² = 18x²
By placing the cubes in a row,
The length of newly formed cuboid (l) = 3x
Breadth (b) = x
and height (h) = x
∴ Surface area of the cuboid so formed

Question 10.
The dimensions of a room are 12.5 m by 9 m by 7 m. There are 2 doors and 4 windows in the room; each door measures 2.5 m by 1.2 m and each window 1.5 m by 1 m. Find the cost of painting the walls at Rs 3.50 per square metre.
Solution:
Dimensions of a room = 12.5 m x 9 m x 7 m
∴ Total surface area of the walls = 2(1 + b) x h = 2(12.5 + 9) x 7 m²
= 2 x 21.5 x 7 = 301 0 m²
Area of 2 doors = 2 x (2.5 x 1.2) m² = 2 x 3.00 = 6 m²
Area of 4 windows = 4 x (1.5 x 1) m²
4 x 1.5 = 6 m²
∴ Remaining area of the walls = 301 -(6 + 6) m²
= 301 – 12 = 289 m²
∴ Rate of painting the walls = Rs 3.50 per m²
∴ Total cost = Rs 3.50 x 289 = Rs 1011.50

Question 11.
A field is 150 m long and 100 m wide. A plot (outside the field) 50 m long and 30 m wide is dug to a depth of 8 m and the earth taken out from the plot is spread evenly in the field. By how much the level of field is raised ?
Solution:
Length of the plot (l) = 50 m
Width (b) = 30 m
and depth (h) = 8 m
∴ Volume of the earth dug out = l x b x h = 50 x 30 x 8 = 12000 m³
Length of the field = 150 m
and breadth = 100 m
∴ Height of the earth spread out on the field

Question 12.
Two cubes, each of volume 512 cm³ are joined end to end, find the surface area of the resulting cuboid.
Solution:
Volume of each cube = 512 cm³

Now by joining the two equal cubes of side 8 cm, the length of so formed cuboid (l)
= 2 x 8 = 16 cm
Breadth (b) = 8 cm
and height (h) = 8 cm
∴ Surface area = 2( l x b + b x h + h x l)
= 2(16 X 8 + 8 X 8 + 8X16) cm²
= 2(128 + 64 + 128) cm²
= 2 x 320 = 640 cm²

Question 13.
Three cubes whose edges measure 3 cm, 4 cm and 5 cm respectively are melted to form a new cube. Find the surface area of the new cube formed.
Solution:
Edge of first cube = 3 cm
∴ Volume = a³ = (3)³ 27 cm³
Edge of second cube = 4 cm
∴Volume = a³ = (4)³ = 64 cm³
Edge of third cube = 5 cm
∴ Volume = a³ = (5)³ = 125 cm³
Volume of three cubes together = 27 + 64+ 125 = 216 cm³
∴ Volume of the new cube = 216 cm³

Question 14.
The cost of preparing the’walls of a room 12 m long at the rate of Rs 1.35 per square metre is Rs 340.20 and the cost of matting the floor at 85 paise per square metre is Rs 91.80. Find the height of the room.
Solution:
Length of the room (l) = 12 m
Rate of matting the floor = 85 paise per m²
Total cost of matting = Rs 91.80

Question 15.
The length of a hall is 18 m and width 12 m. The sum of the areas of the floor and the flat roof is equal to the sum of the areas of the four walls. Find the height of the wall.
Solution:
Length of hall (l) = 18 m
and breadth (b) = 12 m
∴ Area of floor = l x b = 18 x 12 = 216 m²
and area of roof = 216 m²
Total area of floor and roof
= (216 + 216) m² = 432 m²
∴ Area of four walls = 432 m²
But area of 4 walls = 2(l + b) x h
∴ 2h (l + b) = 432
⇒ 2h (18 + 12) = 432
⇒ 2h x 30 = 432 432
⇒ h = 43260 = 7.2m
∴ Height of the wall = 7.2 m

Question 16.
A metal cube of edge 12 cm is melted and formed into three smaller cubes. If the edges of the two smaller cubes are 6 cm and 8 cm, find the edge of the third smaller cube.
Solution:
Edge of metal bigger cube = 12 cm
∴ Volume = (12)³ = 1728 cm³
∴ Sum of volumes of 3 smaller cubes = 1728 cm³
Edge of first smaller cube = 6 cm
∴ Volume = (6)³ = 216 cm³
Edge of second smaller cube = 8 cm
∴ Volume = (8)³ = 512 cm³
Sum of volumes of two smaller cubes = 216+ 512 = 728 cm³
∴ Volume of third smaller cube = 1728-728 cm³ = 1000 cm³

Question 17.
The dimensions of a cinema hall are 100 m, 50 m and 18 m. How many persons can sit in the hall if each person requires 150 m3 of air ?
Solution:
Length of cinema hall (l) = 100 m
Breadth (b) = 50 m
and height (h) = 18 m
∴ Volume of air of the hall = l x b x h
= 100 x 50 x 18 m³
= 90000 m³
Each person requires air = 150 m³
∴ Number of persons = 90000150= 600

Question 18.
The external dimensions of a closed wooden box are 48 cm, 36 cm, 30 cm. The box is made of 1.5 cm thick wood. How many bricks of size 6 cm x 3 cm x 0.75 cm can be put in this box ?
Solution:
Outer dimensions of a closed wooden box = 48 cm x 36 cm x 30 cm
Thickness of wood = 1.5 cm.
∴ Inner length (l) = 48 – 2 x 1.5 cm = 48 – 3 = 45 cm
Breadth(b) = 36-2 x 1.5 = 36-3 = 33 cm
Height (h) = 30 – 2 x 1.5 = 30 – 3 = 27 cm
∴ Volume of inner box = l x b x h = 45 x 33 x 27 cm³ = 40095 cm³
Volume of one brick of size 6 cm x 3 cm x 0.75 cm
= 6 x 3 x 0.75 = 6 x 3 x 34 cm³ = 272 cm³
∴ Number of bricks = 40095×227
= 1485 x 2 = 2970 bricks

Question 19.
The dimensions of a rectangular box are in the ratio of 2 : 3 : 4 and the difference between the cost of covering it with sheet of paper at the rates of Rs 8 and Rs 9.50 per m2 is Rs 1,248. Find the dimensions of the box.
Solution:
Ratio in the dimensions of a box =2:3:4
Difference in total cost = Rs 1,248
Difference in rates = Rs 9.50 – Rs 8 = Rs 1.50
Let length (l) = 2x
Then breadth (b) = 3x
and height (h) = 4x
∴ Surface area = 2 (l x b + b x h + h x l)
= 2(2x 3x  + 3x x 4x + 4x x 2x)
= 2(6x² + 12x² + 8 x²) = 2 x 26x² = 52x²
First rate of paper = Rs 9.50 per m²
and second rate = 8.00 per m²
∴ First cost = Rs 52x² x 9.50
and second cost = Rs 52x² x 8
∴ 52x² x 9.50 – 52x² x 8= 1248
⇒ 52x² (9.50 – 8) = 1248
⇒ 52x²(1.50) = 1248