Surface Area And Volume Of A Cuboid And A Cude
Class 9th Mathematics RD Sharma Solution
- Find the lateral surface area and total surface area of a cuboid of length 80…
- Find the lateral surface area and total surface area of a cube of edge 10 cm.…
- Find the ratio of the total surface area and lateral surface area of a cube.…
- Mary wants to decorate her Christmea tree. She wants to place the tree on a…
- The length, breadth and height of a room are 5 m, 4 m and 3 m respectively.…
- Three equal cubes are placed adjacently in a row. Find the ratio to total…
- A 4 cm cube is cut into 1 cm cubes. Calculate the total surface area of all the…
- The length of a hall is 18 m and the width 12 m. The sum of the areas of the…
- Hameed has built a cubical water tank with lid for his house, with each other…
- Each edge of a cube is increases by 50%. Find the percentage increase in the…
- The dimensions of a rectangular box are in the ratio of 2 : 3 : 4 and the…
- A closed iron tank 12 m long, 9 m wide and 4 m deep is to be made. Determine…
- Ravish wanted to make a temporary shelter for his car by making a box-like…
- An open box is made of wood 3 cm thick. Its external length, breadth and…
- The cost of preparing the walls of a room 12 m long at the rate of Rs. 1.35…
- The dimensions of a room are 12.5 m by 9 m by 7 m. There are 2 doors and 4…
- The length and breadth of a hall are in the ratio 4 : 3 and its height is 5.5…
- A wooden bookshelf has external dimensions as follows : Height = 110 cm, Depth…
- The paint in a certain container is sufficient to paint on area equal to 9.375…
- A cuboidal water tank is 6 m long, 5 m wide and 4.5 m deep. How many litres of…
- A cubical vessel is 10 m long and 8 m wide. How high must it be made to hold…
- Find the cost of digging a cuboidal pit 8 m long, 6 m broad and 3 m deep at the…
- If V is the volumw of cuboid of dimensions a,b,c and S is its surface area,…
- The areas of three adjacent faces of a cuboid are x, y, and z. If the volumes…
- If the areas of three adjacent faces of a cuboid are 8 cm^2 , 18 cm^2 and 25…
- The breadth of a room is twice its height, one half of its length and the…
- A river 3 m deep and 40 m wide is flowing at the rate of 2 km per hour. How…
- Water in a canal 30 dm wide and 12 dm deep, is flowing with a velocity of 100…
- Three metal cubes with edges 6 cm, 8 cm and 10 cm respectively are melted…
- Two cubes, each of volume 512 cm^3 are joined end to end. Find the surface…
- Half cubic metre of gold-sheet is extended by hammering so as to cover an area…
- A metal cube of edge 12 cm is melted and formed into three smaller cubes. If…
- The dimensions of a cinema hall are 100 m, 50 m and 18 m. How many persons can…
- Given that 1 cubic cm of marble weighs 0.25 kg, the weight of marble block 28…
- A box with lid is made of 2 cm thick wood. Its external length, breadth and…
- The external dimensions of a closed wooden box are 48 cm, 36 cm, 30 cm. The…
- How many cubic centimetres of iron are there in an open box whose external…
- A cube of 9 cm egde is immersed completely in a rectangular vessel containing…
- A rectangular container, whose base is a square of side 5 cm, stands on a…
- A field is 200 m long and 150 m broad. There is a plot, 50 m long and 40 m…
- A field is in the form of a rectangle of length 18 m and width 15 m. A pit,…
- A rectangular tank is 80 m long and 25 m broad. Water flows into it through a…
- Water in a rectangular reservoir having base 80 m by 60 m is 6.5 m deep. In…
- A village having a population of 4000 requires 150 litres of water per head…
- A child playing with building blocks, which are of the shape of the cubes, has…
- A godown measures 40 m25 m10 m. Find the maximum number of wooden crates each…
- A wall of length 10 m was to be built across an open ground. The height of the…
- If two cubes each of side 6 cm are joined face to face, then find the volume of the…
- If A1, A2 and A3 denote the areas of three adjacent faces of a cuboid, then its volume…
- Three cubes of metal whose edges are in the ratio 3 : 4 : 5 are melted down into a…
- The length of the longest rod that can be fitted in a cuboid vessel of edge 10 cm long,…
- If the perimeter of each face of a cube is 32 cm, find its lateral surface area. Note…
- If l is the length of a diagonal of a cube of volume V, thenA. 3V=l^3 B. root 3 V=l3 C.…
- Find the edge of a cube whose surface area is 432 m^2 .
- Three equal cubes are placed adjacently in a row. The ratio of the total surface area…
- If V is the volume of a cuboid of dimensions x, y, z and A is its surface area, then…
- A cuboid has total surface area of 372 cm^2 and its lateral surface area is 180 cm^2 ,…
- The sum of the length, breadth and depth of a cuboid is 19 cm and its diagonal is 5…
- Three cubes of each side 4 cm are joined end to end. Find the surface area of the…
- If the length of a diagonal of a cube is 8 root 3 cm, then its surface area isA. 512…
- The surface area of a cuboid is 1300 cm^2 . If its breadth is 10 cm and height is 20…
- If each edge of a cube is increased by 50%, the percentage increase in its surface area…
- If the volumes of two cubes are in the ratio 8 : 1, then the ratio of their edges isA.…
- The volume of a cube whose surface area os 96 cm^2 , isA. 16 root 2 cm^3 B. 32 cm^3 C.…
- The length, width and height of a rectangular solid are in the ratio of 3 : 2 : 1. If…
- A cube whose volume is 1/8 cubic centimetre is placed on top of a cube whose volume is…
- If the areas of the adjacent faces of a rectangular block are in the ratio 2 : 3 : 4…
- If each edge of a cube, of volume V, is doubled, then the volume of the new cube isA.…
- If each edge of a cuboid of surface area S is doubled, then surface area of the new…
- The area of the floor of a room is 15 m^2 . If its height is 4 m, then the volume of…
- The cost of constructing a wall 8 m long, 4 m high and 20 cm thick at the rate of Rs.…
- 10 cubic metres clay is uniformly spread on a land of area 10 ares. The rise in the…
- Volumeof a cuboid is 12 cm^3 . The volume (in cm^3) of a cuboid whose sides are double…
- If the sum of all the edges of acube is 36 cm, then the volume (in cm^3) of that cube…
- The number of cubes of side 3 cm that can be cut from a cuboid of dimensions 10 cm9…
- On a particular day, the rain fall recorded in a terrace 6 m long and 5 m board is 15…
Exercise 18.1
Question 1.Find the lateral surface area and total surface area of a cuboid of length 80 cm, breadth 40 cm and height 20 cm.
Answer:
Given,
Length of a cuboid =80 cm
Breadth of cuboid = 40 cm
Height of cuboid = 20 cm
So, lateral surface area of cuboid = 2(l+b)h = 2(80+40)×20 = 4800 cm2
Total surface area of cuboid = 2(lb+bh+hl) = 2(80×40+40×20+20×80) = 11200 cm2
Question 2.
Find the lateral surface area and total surface area of a cube of edge 10 cm.
Answer:
Given,
Edge of cube = 10 cm
So, total surface area of cube = 6a2 = 6 ×100 = 600cm2
Lateral surface area of cube = 4a2 = 4×100 = 400 cm2
Question 3.
Find the ratio of the total surface area and lateral surface area of a cube.
Answer:
Question 4.
Mary wants to decorate her Christmea tree. She wants to place the tree on a wooden block covered with coloured paper with picture of Sants Claus on it. She must know the exact quantity of paper to buy for this purpose. If the box has length, breadth and height as 80 cm, 40 cm and 20 cm respectively. How many square sheets of paper of side 40 cm would she require?
Answer:
Given,
Dimensions of box = 80cm ×40cm×20cm
So,
quantity of paper required by Mary to cover the wooden block is total surface area = 2(80×40+40×20+20×80)
= 11200 cm2
Hence,
number of square sheet of side 40 cm required =
= 
Question 5.
The length, breadth and height of a room are 5 m, 4 m and 3 m respectively. Find the cost of white washing the walls of the room and the ceiling at the rate of Rs. 7.50 m2.
Answer:
Given,
Dimension of room = 5m×4m×3m
Area of walls = 2(l+b)h = 2×9×3 = 54 m2
Area of ceiling = 20m2
So, total area to be whitewashed = 20+54 = 74m2
Hence, cost of whitewashing at rate Rs 7.50 per m2 = 7.50×74 = Rs.555
Question 6.
Three equal cubes are placed adjacently in a row. Find the ratio to total surface area on the new cuboid to that of the sum of the surface areas of the three cubes.
Answer:
Given,
Let length of an edge of three equal cubes = a
Total surface area of 3 cubes = 3× 6a2 = 18a2
(While putting these cubes in a row adjacently, a cuboid is formed)
∴ length of cuboid = 3a
Breadth of cuboid = a
Height of cuboid = a
So,
total surface area of cuboid = 2(3a2+a2+3a2) = 14 a2
Hence, 
Question 7.
A 4 cm cube is cut into 1 cm cubes. Calculate the total surface area of all the small cubes.
Answer:
Given,
Original edge of cube = 4 cm
Original volume of cube = 43 = 64 cm3
After cutting ,
Edge of cube = 1 cm
Hence volume of new cube = 13 = 1 cm3
Number of small cubes thus formed = 
Hence , total surface area of 64 cubes = 64× 6a2 = 64×6 = 384 m2
Question 8.
The length of a hall is 18 m and the 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 hall.
Answer:
Given,
Length of a hall = 18 m
Width of hall = 12 m
Sum of area of floor and area of roof = sum of area of four walls
Let height of hall = h m
= l×b+l×b = 2(l+b)h
= 2lb = 2(l+b)h = h = 
Question 9.
Hameed has built a cubical water tank with lid for his house, with each other edge 1.5 m long. He gets the outer surface of the tank excluding the base, covered with square tiles of side 25 cm. Find how much he would spend for the tiles, if the cost of tiles is Rs. 360 per dpzen.
Answer:
Given,
Each edge of cubical water tank = 1.5m
So,
area of 5 faces of tank(excluding base) = 5×(1.5)2 = 5×150×150 cm2
Area of each tile = 25×25 = 625 cm2
Number of tiles required = 
∵ cost of 1 dozen tiles = Rs.360
∴ cost of 180 tiles (15 dozen) = 15×360 = Rs. 5400
Question 10.
Each edge of a cube is increases by 50%. Find the percentage increase in the surface of area of the cube.
Answer:
Given,
Increment in each edge = 50%
Let edge of cube = a
So surface area of cube = a2
(after 50% increase )
Length of edge of cube = a+ a×
Surface area of cube = 
Increment in area = 
% increment = 
Question 11.
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. 1248. Find the dimensions of the box.
Answer:
Given,
Ratio of dimensions of rectangular box = 2:3:4
Let dimensions are = 2x ,3x , 4x
So, total surface area of box = 2(6x2+12x2+8x2) = 52 x2
Let, c1 = cost of covering at rate Rs. 8 per m2
C1 = 8×52x2
Let c2 = cost of covering at rate Rs.9.50 per m2
C2 = 9.50× 52x2
= C2 – C1 = 1248 (given)
= 52x2×9.50 – 52x2×8 = 1248
= 52x2 = 
= x2 = 
= x = 4
Hence dimensions of box are = 2x = 2×4 = 8 cm
= 3x = 3×4 = 12 cm
=4x= 4×4 = 16cm
Question 12.
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.
Answer:
Given,
Length of closed iron tank = 12m
Breadth = 9m
Height = 4m
Total surface area of tank = 2(12×9+9×4+4×12) = 2(192) = 384 m2
Width of iron sheet = 2m (given)
So, length of iron sheet = 
Hence, cost of iron sheet at rate Rs.5 per meter = 5×192 = Rs.960
Question 13.
Ravish wanted to make a temporary shelter for his car by making a box-like structure with tarpaulin that covers all the four sides and the top of the car (with the front face as a flap which can be rolled up). Assuming that the stitching margins are very small, and therefore negligible, how much tarpaulin would be required to make the shelter of height 2.5 m with base dimensions 4 m × 3 m?
Answer:
Given,
Height of shelter = 2.5 m
Base dimensions = l×b = 4m×3m
So,
area of tarapaulin needed = lateral surface area of shelter +area of roof
= 2(l+b)h + l×b
= 2(4+3)2.5 +12 = 47 m2
Question 14.
An open box is made of wood 3 cm thick. Its external length, breadth and height are 1.48 m, 1.16 m and 8.3 dm. Find the cost of painting the inner surface of Rs. 50per sq. metre.
Answer:
Given,
Thickness of wood = 3cm
External length of box = 1.48 m = 148 cm
External breadth of box = 1.16 m = 116 cm
External height of box = 8.3 dm = 83 cm
So, inner dimensions of box = external dimension – thickness of wood
Inner length = 148- (2×3) = 142 cm
Inner breadth = 116 –(2×3) = 110 cm
Inner height = 83-3 = 80 cm
Hence , inner surface area of box = lateral surface area +area of roof
=2(l+b)h + l×b
= 2 (142+110)83 + 142×110 = 55940 cm2 = 5.5940 m2
∴ cost of painting at rate Rs.50 per m2 = 50×5.5940 = Rs. 279.70
Question 15.
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.
Answer:
Given,
Length of room = 12m
Let breadth of room = b m
Let height of room = h m
So, lateral surface area of room = 2(l+b)h m2
Area of floor = l×b m2
Cost of matting the floor = area of floor × rate of matting
= 91.80 = l×b ×.85
= b = 
Cost of preparing wall = area of four walls ×rate of preparing
= 340.20 = 2(l+b)h ×
= 
Question 16.
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.
Answer:
Given,
Dimension of room = 12.5m×9m×7m
Dimension of door = 2.5m×1.2m
Dimension of window = 1.5m×1m
So, area of room = 2(l+b)h = 2(12.5+9)×7 = 301 m2
Area of 2 doors = 2(2.5×1.2) = 6m2
Area of 4 windows = 4(1.5×1) = 6 m2
So, area to be painted = (301-12) = 289 m2
∴ cost of painting walls at rate Rs.3.50 per m2 = 3.50×289 = Rs.1011.50
Question 17.
The length and breadth of a hall are in the ratio 4 : 3 and its height is 5.5 metres. The cost of decorating its walls (including doors and windows) at Rs. 6.60 per square metre is Rs. 5082. Find the length and breadth of the room.
Answer:
Given,
Ratio of length and breadth of a hall = 4:3
Let length of hall = 4x m
Let breadth of hall = 3x m
Height of hall = 5.5m (given)
So, lateral surface area of hall = 2(l+b)h =2(7x) ×5.5 = 77x m2
Cost of decorating walls = lateral surface area +rate of decoration per m2
= 5082 = 
= 
So, length of hall = 4x = 4×10 = 40 m
Breadth of hall = 3x = 3×10 = 30 m
Question 18.
A wooden bookshelf has external dimensions as follows : Height = 110 cm, Depth = 25 cm, Breadth = 85 cm (See Fig. 18.5). The thickness of the plank is 5 cm everywhere. The external faces are to be polished and the inner faces are to be painted. If the rate of polishing is 20 paise per cm2 and the rate of painting is 10 paise per cm2. Find the total expenses required for polishing and painting the surface of the bookshelf.
Answer:
Given,
External dimensions of wooden shelf = 85cm×25cm×110cm
Thickness of plank = 5 cm
So,
area to be polished = area of four walls +area of back+ area of front beading
= [2(110+85)×25 +110×85 +(75×5)×4] = 21700 cm2
Cost of polishing at rate 20paise per cm2 = 
Internal surface area = area of five faces of 3 cuboids each of dimension (75cm×30cm×20cm)
Total surface area of 3 cuboids = 3[2(75×30+30×20+20×75)] – 3(75×30)
= 6(2250+600+1500) – 6750 = 19350 cm2
∴ cost of painting at rate 10p per cm2 = 
Hence, total expense = Rs.(4340+1935) = Rs. 6275
Question 19.
The paint in a certain container is sufficient to paint on area equal to 9.375 m2. How many bricks of dimension 22.5 cm×10cm ×7.5 cm can be painted out of this container?
Answer:
Given,
Dimension of a brick = 22.5 cm×10cm×7.5 cm
Total surface area = 2(lb + bh + hl)
= 2[22.5× 10 + 10× 7.5 + 22.5 × 7.5] cm2
= 2(225 + 75 + 168.75) cm2
= 2(468.75) cm2
= 937.5 cm2
Area that can be painted = 9.375 m2
As 1 m2 = 10000 cm2
9.375 m2 = 9.375 × 10000
= 93750 cm2
Number of bricks = 
= 100 bricks
Exercise 18.2
Question 1.A cuboidal water tank is 6 m long, 5 m wide and 4.5 m deep. How many litres of water can it hold?
Answer:
Given,
Length of cuboidal tank = 6m
Breadth of cuboidal tank = 5 m
Depth of cuboidal tank = 4.5 m
So, capacity of cuboid = l×b×h = 6×5×4.5 = 135m3= 135000 litre
Question 2.
A cubical vessel is 10 m long and 8 m wide. How high must it be made to hold 380 cubic metres of a liquid?
Answer:
Given,
Length of cuboidal vessel = 10m
Width of cuboidal vessel = 8m
Let height of vessel = h m
So,
volume of vessel = l×b×h = 10×8×h m3
= 10×8×h = 380
= h = 
Question 3.
Find the cost of digging a cuboidal pit 8 m long, 6 m broad and 3 m deep at the rate of Rs. 30 per m3.
Answer:
Given,
Dimension of cuboidal pit = 8m×6m×3m
So, volume of pit = 8×6×3 = 144 m3
Cost of digging pit at rate Rs.30 per m3 = 30×144 = Rs.4320
Question 4.
If V is the volumw of cuboid of dimensions a,b,c and S is its surface area, then prove that
Answer:
Given,
V= volume of cuboid
S = surface area of cuboid
=a = length of cuboid
=b =breadth of cuboid
= c = height of cuboid
V = abc
S = 2(ab+bc+ca)
Dividing surface area by volume
= 
= 
Question 5.
The areas of three adjacent faces of a cuboid are x, y, and z. If the volumes is V, prove that V2 = xyz.
Answer:
Given,
Area of 3 adjacent faces of a cuboid = x, y, z
V = volume of cuboid
Let , a,b,c are respectively length , breadth, height of each faces of cuboid
So, x = ab
= y = bc
= z = ca
V = abc
Hence , xyz = ab×bc×ca = (abc)2 = v2 (v=abc)
= v2 = xyz Proved.
Question 6.
If the areas of three adjacent faces of a cuboid are 8 cm2, 18 cm2 and 25 cm3. Find the vlume of the cuboid.
Answer:
Given,
Area of 3 adjacent faces of cuboid = x= 8 cm2 , y = 18 cm2 , z = 25 cm2
From previous question we get ,
= v2 =xyz
= v2= 8×18×25 = 3600
=v =
Question 7.
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.
Answer:
Given,
Breadth of room = 2×height of room = b = 2×h = h = 
Breadth of room = 
Volume of room = 512 dm3
= l×b×h = 512
= 2b×b×
= b3 = 512 = b = 
So, breadth of room = 8 dm
Length of room = 2b = 2×8 = 16 dm
Height of room = 
Question 8.
A river 3 m deep and 40 m wide is flowing at the rate of 2 km per hour. How much water will fall into the sea ina minute?
Answer:
Given,
Depth of river = 3m
Width of river = 40 m
Rate of flow of river = 20km/h
Length of river = distance travelled by water in 1 minute at speed 2km/h
=
So, volume of water = 
Question 9.
Water in a canal 30 dm wide and 12 dm deep, is flowing with a velocity of 100 km per hour. How much area will it irrigate in 30 minutes if 8 cm of standing water is desired?
Answer:
Given,
Width of canal = 30 dm = 3m
Depth of canal = 12 dm = 1.2 m
Velocity of flow = 100 km/h
Length of cuboid thus formed = distance travelled by water in 30 minute at speed 100km/h
=
Volume of water = 50000×3×1.2 m3
Water accumulated in field forms a cuboid of base area equals to area of field and height = 
= hence , area of field = 
Question 10.
Three metal cubes with edges 6 cm, 8 cm and 10 cm respectively are melted together and formed into a single cube. Find the volume, surface area and diagonal of the new cube.
Answer:
Given,
Edges of 3 metal cubes = 6cm, 8cm, 10 cm
Volume of these cubes together = 63+83+103 = 216+512+1000 = 1728 cm3
Let volume of new cube formed = V
So, V = 1728
= a3 = 1728 = a =
Surface area of new cube = 6a2 = 6×122 = 864 cm2
Diagonal of new cube = 
Question 11.
Two cubes, each of volume 512 cm3 are joined end to end. Find the surface area of the resulting cuboid.
Answer:
Given,
Volume of 2 equal cubes = 512 cm3
= a3 = 512 = a = 
When these cubes joined end to end a cuboid is formed so,
Length of cuboid = 8+8 = 16 cm
Breadth of cuboid = 8 cm
Height of cuboid = 8 cm
So, surface area of cuboid = 2(lb+bh+hl) =2(16×8+8×8+8×16)
= 2×320 = 640 cm2
Question 12.
Half cubic metre of gold-sheet is extended by hammering so as to cover an area of 1 hectare. Find the thickness of the gold-sheet.
Answer:
Given,
Volume of the gold sheet = 
Area of sheet = 1 hectare = 10000 m2
So, thickness of gold sheet = 
Question 13.
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.
Answer:
Given,
Edge of metal cube = 12 cm
Volume of cube = a3 = 123 = 1728 cm3
Edge of 2 smaller cube = 6cm , 8 cm
Let edge of 3rd cube = x cm
So, 63+83+ x3 = 1728
= x3 = 1728 – (216+512) = 1000
= x = 
Question 14.
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?
Answer:
Given,
Dimension of cinema hall = 100m×50m×18m
Volume of cinema hall = 100×50×18 m3
Number of persons can sit in cinema hall = 
=
Question 15.
Given that 1 cubic cm of marble weighs 0.25 kg, the weight of marble block 28 cm in width and 5 cm thick is 112 kg. Find the length of the block.
Answer:
Given,
Weight of 1m3 marble = 0.25 kg
Weight of marble 28 cm wide and 5 cm thick = 112 kg
Let length of block = l cm
Then volume of block = l×28×5 cm3 = 140 l cm3
Weight of block = 140 l ×0.25
= 112 = 140 l ×0.25
= l = 
Question 16.
A box with lid is made of 2 cm thick wood. Its external length, breadth and height are 25 cm, 18 cm and 15 cm respectively. How much cubic cm of liquid can be places in it> Also, find the volume of the wood used in it.
Answer:
Given,
External dimensions of box = 25cm×18cm×15cm
Thickness of wood = 2 cm
So, internal dimensions of box = (25-4) cm×(18-4) cm×(15-4) cm
= 21cm×14cm×11cm
Volume of liquid can place in it = 21 ×14×11 = 3234 cm3
Volume of wood use in it = external volume – internal volume
= 
Question 17.
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 × 3 cm×0.75 cm can be put in this box?
Answer:
Given,
External dimensions of closed wooden box = 48cm×36cm×30cm
Thickness of wood = 1.5 cm
So inner dimensions of box would be = [(48-3)×(36-3)×(30-3)]
= 45cm×33cm×27cm
Volume of box = 45×33×27 = 38880 cm3
Volume of a brick = 6×3×0.75 = 13.5 cm3
So, number of bricks can put in box = 
Question 18.
How many cubic centimetres of iron are there in an open box whose external dimension are 36 cm, 25 cm and 16.5 cm, the iron being 1.5 cm thick throughout? If 1 cubic cm of iron weighs 15 g, find the weight of the empty box in kg.
Answer:
Given,
External dimensions of box = 36cm×25cm×16.5cm
Thickness of iron = 1.5 cm
Inner dimensions of box = [(36-3)×(25-3)×(16.5-1.5)]
=33cm×22cm×15cm
Volume of iron used = external volume – internal volume
= 36×25×16.5 – 33×22×15 = 3960 cm3
∵ weight of 1 cm3 iron = 15 gm (given)
∴ weight of 3960 cm3 iron = 3960×15 = 59400 gm = 59.4 kg
Question 19.
A cube of 9 cm egde is immersed completely in a rectangular vessel containing water. If the dimensions of the base are 15 cm and 12 cm, find the rise in water level in the vessel.
Answer:
Given,
Edge of cube = 9 cm
Dimension of base of rectangular vessel = 15cm×12cm
Volume of cube = 9×9×9 = 729 cm3
Area of base of vessel = 15×12 = 180 cm2
So, rise of water in vessel = 
Question 20.
A rectangular container, whose base is a square of side 5 cm, stands on a horizontal table, and holds water upto 1 cm from the top. When a cube is placed in the water it is completely submerged, the water rises to the top and 2 cubic cm of water overflows. Calculate the volume of the cube and also the length of its edge.
Answer:
Given,
Side of square base of container = 5 cm
Height of water level in container = 1 cm from top
Water overflow after submerging of cube = 2 cm3
Let edge of cube = a cm
Volume of cube = a3 cm3
= a3 = 52×1 +2 = 27 cm3
= a 
Question 21.
A field is 200 m long and 150 m broad. There is a plot, 50 m long and 40 m broad, near the field. The plot is dug 7 m deep and the earth taken out is spread evenly on the field. By how many metres is the level of the field raised? Give the answer to the second place of decimal.
Answer:
Given,
Length of field = 200 m
Breadth of field = 150 m
Dimension of plot = 50m×40m
Depth up to which plot is dug = 7 m
Volume of earth dug out = 50×40×7 = 14000 m3
Let level of earth rises in field = h meter
Hence,
= 200×150×h = 14000
= h =
Question 22.
A field is in the form of a rectangle of length 18 m and width 15 m. A pit, 7.5 m long, 6 m broad and 0.8 m deep, is dug in a corner of the field and the earth taken out is spread over the remaining area of the field. Find out the extent to which the level of the field has been raised.
Answer:
Given,
Dimension of rectangular field = 18m ×15m
Dimension of pit = 7.5m×6m×0.8m
So, area on which earth taken out is to be spread = 18×15 – 7.5×6=225 m2
Let level of earth rises in field = h meter
So,
=225×h = 7.5×6×0.8
= h = 
Question 23.
A rectangular tank is 80 m long and 25 m broad. Water flows into it through a pipe whose cross-section is 25 cm2, at the rate of 16 km per hour. How much the level of the water rise in the tank in 45 minutes.
Answer:
Given,
Dimension of rectangular tank = 80m×25m = 8000cm×2500cm
Cross section area of pipe = 25 cm2
Rate of flow of water through pipe = 16 km/h
Length of pipe = flow of water through it in 45 minute at speed 16 km/h
= 
Volume of pipe = 
Let level of water rises in tank in 45 minute = h cm
= 8000×2500×h = 
= h = 
Question 24.
Water in a rectangular reservoir having base 80 m by 60 m is 6.5 m deep. In what time can the water be emptied by a pipe of which the cross-section is a square of side 20 cm, if the water runs through the pipe at the rate of 15 km/hr.
Answer:
Given,
Dimension of rectangular reservoir = 80m×60m
Depth of water in reservoir = 6.5 m
Side of square pipe = 20cm
Rate of flow of water through pipe = 15 km/h
Volume of reservoir = 80×60×6.5 m3
Volume of pipe = 
Let reservoir be emptied in t hours
= 
= t =
Question 25.
A village having a population of 4000 requires 150 litres of water per head per day. It has a tank measuring 20 m×15 m×6 m. For how many days will the water of this tank last?
Answer:
Given,
Population of village = 4000
Requirement of water per head per day = 150 litre
Dimension of tank = 20m×15m×6m
Let the water in tank last for x days ..
= 
= 
Question 26.
A child playing with building blocks, which are of the shape of the cubes, has built a structure as shown in Fig. 18.12. If the edge of each cube is 3 cm, find the volume of the structure built by the child.
Answer:
Given,
Edge of cube = 3 cm
Volume of cube = 33 = 27 cm3
Number of cubes = 15
Volume of structure = number of cubes × volume of one cube
= 15×27 = 405 cm3
Question 27.
A godown measures 40 m×25 m×10 m. Find the maximum number of wooden crates each measuring 1.5 m×1.25 m×0.5 m that can be stored in the godown.
Answer:
Given,
Dimension of godown = 40m×25m×10m
Dimension of wooden crates = 1.5m×1.25m×0.5m
Number of wooden crates = 
Question 28.
A wall of length 10 m was to be built across an open ground. The height of the wall is 4 m and thickness of the wall is 24 cm. If this wall is to be built up with bricks whose dimensions are 24 cm×12 cm×8 cm, how many bricks would be required?
Answer:
Given,
Length of wall = 10m
Height of wall = 4m
Thickness of wall = 24 cm
Dimension of bricks = 24cm×12cm×8cm
So, number of bricks required = 
Cce - Formative Assessment
Question 1.If two cubes each of side 6 cm are joined face to face, then find the volume of the resulting cuboid.
Answer:
Given,
Side of two equal cubes = 6 cm
(after joining them face to face a cuboid is formed)
Length of cuboid = 6+6 = 12 cm
Breadth of cuboid = 6 cm
Height of cuboid = 6 cm
So, volume of cuboid = l×b×h = 12×6×6 = 432 cm3
Question 2.
If A1, A2 and A3 denote the areas of three adjacent faces of a cuboid, then its volume is
A. A1A2A3
B. 2 A1A2A3
C. 
D. 
Answer:
Given,
A1A2,A3 = areas of 3 adjacent faces of cuboid
V = volume of cuboid
Let dimensions of cuboid = 
A1 = 
A2 = 
A3 = 
V = 
So, A1A2A3 = 
= 
Question 3.
Three cubes of metal whose edges are in the ratio 3 : 4 : 5 are melted down into a single cube whose diagonal is 12
cm. Find the edges of three cubes.
Answer:
Given,
Ratio of edge of 3 cubes = 3:4:5
Diagonal of new cube formed = 12√3 cm
Let edges are = 3x ,4x 5x
Comined volume = 
= 216 x3
New diagonal = √3a = 12√3 = a = 12 cm
So, a3 = 216 x3
= 12×12×12 = 216 x3
= 
=x =
So , edges are = 3x = 3×2 = 6 cm
=4x = 4×2 = 8cm
= 5x = 5×2 = 10 cm
Question 4.
The length of the longest rod that can be fitted in a cuboid vessel of edge 10 cm long, is
A. 10 cm
B. 10
cm
C. 10 cm
D. 20 cm
Answer:
Given,
Edge of cube = 10 cm
Longest rod that can be fitted in cube = diagonal of cube = 
= 
cm
Question 5.
If the perimeter of each face of a cube is 32 cm, find its lateral surface area. Note that four faces which meet the base of a cube are called its leteral faces.
Answer:
Given,
Perimeter of each face of a cube = 32 cm
Let each edge will be = a cm
So,
4a = 32
= a = 
Lateral surface area of cube = 4a2 = 4×82 = 256 cm2
Question 6.
If l is the length of a diagonal of a cube of volume V, then
A. 3V=l3
B. V=l3
C. 3V=2 l3
D. 3V= l3
Answer:
Given ,
= 
= v = volume of cube
So, 
= 
∵ v = a3
= v = 
=
Question 7.
Find the edge of a cube whose surface area is 432 m2.
Answer:
Given,
Surface area of cube = 432 m2
Let edge of cube = a m
= 6 a2 = 432
= a2 = 
= a = 
Question 8.
Three equal cubes are placed adjacently in a row. The ratio of the total surface area of the resulting cuboid to that of the sum of the surface areas of three cubes, is
A. 7 : 9
B. 49 : 81
C. 9 : 7
D. 27 : 23
Answer:
Given,
Three equal cubes are placed adjacently in a row
Let edge of cubes = a
Hence , sum of surface area of 3 cubes = 3×6a2 = 18 a2
So, length of cuboid thus formed = a+a+a = 3a
Breadth of cuboid = a
Height of cuboid = a
Total surface area of cuboid = 
Hence , 
Question 9.
If V is the volume of a cuboid of dimensions x, y, z and A is its surface area, then A/V
A. x2y2z2
B. 
C. 
D. 
Answer:
Volume of a Cuboid = Length × Breadth × Height
Hence, Volume of a Cuboid V = xyz
Surface Area of a Cuboid = 2 × (Length × Breadth + Breadth × Height + Length × Height)
∴ Surface Area of a Cuboid A = 2 × (xy + yz + zx)
Question 10.
A cuboid has total surface area of 372 cm2 and its lateral surface area is 180 cm2, find the area of its base.
Answer:
Given,
Total surface area of cuboid = 372 cm2
Lateral surface area = 180 cm2
= 2(l+b) h = 180
= lh +bh = 
………………(i)
And ,
= 2(lb+bh+hl) = 372
= lb+bh+hl = 
………………(ii)
Putting value of lh+bh from equation (i) to (ii)
= lb +90 = 186
= lb = 186-90 = 96
=l×b = area of base = 96 cm2
Question 11.
The sum of the length, breadth and depth of a cuboid is 19 cm and its diagonal is 5 
cm. Its surface area is
A. 361 cm2
B. 125 cm2
C. 236 cm2
D. 486 cm2
Answer:
Given,
Sum of length ,breadth, height of cuboid = 19 cm
Diagonal of cuboid = 5√5 cm
= 
(Squaring both side )
= 
∵ 
Putting values …
= 192 = 125 + 2
= 
= surface area of cuboid = 236 cm2
Question 12.
Three cubes of each side 4 cm are joined end to end. Find the surface area of the resulting cuboid.
Answer:
Given,
Edge of each cube = 4cm
After joining 3 cubes end to end a cuboid is formed
Length of cuboid = 4+4+4 = 12 cm
Breadth of cuboid = 4 cm
Height of cuboid = 4cm
Surface area of cuboid = 2(lb+bh+hl)
= 2(48+16+48)
= 224 cm2
Question 13.
If the length of a diagonal of a cube is 8 cm, then its surface area is
A. 512 cm2
B. 384 cm2
C. 192 cm2
D. 768 cm2
Answer:
Given,
Length of diagonal of cube = 8√3 cm
∵ √3 a = 8√3
= 
= surface area of cube = 6 a2 = 6× 8×8 = 384 cm2
Question 14.
The surface area of a cuboid is 1300 cm2. If its breadth is 10 cm and height is 20 cm2, find its length.
Answer:
Given,
Surface area of cuboid = 1300 cm2
Breadth of cuboid = 10 cm
Height of cuboid = 20 cm
Let length of cuboid = 
= 
=
=
=
Question 15.
If each edge of a cube is increased by 50%, the percentage increase in its surface area is
A. 50%
B. 75%
C. 100%
D. 125%
Answer:
Given,
Increment in each edge of a cube = 50%
Let original length of edge of cube = a
Original surface area of cube = 6 a2
After increment ,
New length of edge = a + a
New surface area of cube = 6a2 = 6× 
Increase in area = 
% increase in area = 
Question 16.
If the volumes of two cubes are in the ratio 8 : 1, then the ratio of their edges is
A. 8 : 1
B. 2 : 1
C. 2 : 1
D. none of these
Answer:
Given,
Ratio of volumes of two cubes = 
∴ 
= 
Question 17.
The volume of a cube whose surface area os 96 cm2, is
A. 16 
cm3
B. 32 cm3
C. 64 cm3
D. 216 cm3
Answer:
Given,
Surface area of cube = 96 cm2
= 6a2 = 96
= 
= a = 
Volume of cube = a3 = 4×4×4 = 64 cm3
Question 18.
The length, width and height of a rectangular solid are in the ratio of 3 : 2 : 1. If the volume of the box is 48 cm3, the total surface area of the box is
A. 27 cm2
B. 32 cm2
C. 64 cm2
D. 88 cm2
Answer:
Given,
Ratio of length , width and height of rectangular solid = 3:2:1
Volume of box = 48 cm3
Let length of box = 3x cm
Breadth of box = 2x cm
Height of box = x cm
Volume of box = v = 
= 6x3 = 48
= 
= 
Hence . length = 3x = 3×2 = 6 cm
= bredth = 2x = 2×2 = 4 cm
= height = x = 2 cm
Total surface area of box = 
Question 19.
A cube whose volume is 1/8 cubic centimetre is placed on top of a cube whose volume is 1 cm3. The two cubes are then placed on top of a third cube whose volume is 8 cm3. The height of the stacked cubes is
A. 3.5 cm
B. 3 cm
C. 7 cm
D. none of these
Answer:
Given,
Volume of 1st cube = 
Volume of second cube = 1 cm3
Volume of third cube = 8 cm3
∴edge of first cube = a1 = 
∴ edge of second cube = a2 = 
∴ edge of third cube = a3 = 
Hence height of cubes together = a1+a2+a3 = 
Question 20.
If the areas of the adjacent faces of a rectangular block are in the ratio 2 : 3 : 4 and its volume is 9000 cm3, then the length of the shortest edge is
A. 30 cm
B. 20 cm
C. 15 cm
D. 10 cm
Answer:
Given,
Ratio of areas of adjacent faces of cube = 2:3:4
Volume of block = 9000 cm3
= A1:A2:A3 = 2:3:4
= bh:lb:lh = 2:3:4
= b:l = 2:3
=h:l = 2:4
= h:b = 3:4 and v = lbh
Assume that , l = 6x , b= 4x , h = 3x
= 
= 
= x = 
So, smallest edge would be 3x = 3×5 = 15 cm
Question 21.
If each edge of a cube, of volume V, is doubled, then the volume of the new cube is
A. 2 V
B. 4 V
C. 6 V
D. 8 V
Answer:
Given,
Let original edge of cube = a
Then original volume of cube = a3
New edge of cube = 2a
New volume of cube = (2a)3 = 8a3 = 8v ∵v=a3
Question 22.
If each edge of a cuboid of surface area S is doubled, then surface area of the new cuboid is
A. 2 S
B. 4 S
C. 6 S
D. 8 S
Answer:
Given,
Let edges of cuboid = l , b , h
Surface area of cuboid = 
New edges of cuboid = 2l , 2b , 2h
New surface area of cuboid = 
= 4S
Question 23.
The area of the floor of a room is 15 m2. If its height is 4 m, then the volume of the air contained in the room is
A. 60 dm3
B. 600 dm3
C. 6000 dm3
D. 60000 dm3
Answer:
Given,
Area of floor of room = 15 m2
Height of room = 4 m
Volume of air contained in room = area of floor ×height of room
= 15×4 = 60 m3 = 60000 dm3
Question 24.
The cost of constructing a wall 8 m long, 4 m high and 20 cm thick at the rate of Rs. 25 per m3 is
A. Rs. 16
B. Rs. 80
C. Rs. 160
D. Rs. 320
Answer:
Given,
Length of wall = 8m
Breadth of wall = 4 m
Thickness of wall = 20 cm = 
Volume of wall = 
∵ cost of construction of 1m3 wall = Rs. 25
∴ cost of construction of 
Question 25.
10 cubic metres clay is uniformly spread on a land of area 10 ares. The rise in the level of the ground is
A. 1 cm
B. 10 cm
C. 100 cm
D. 1000 cm
Answer:
Given,
Volume of clay = 10 m3
Area of earth = 10 acre = 1000000 cm2
Rise in height of earth level = 
Question 26.
Volumeof a cuboid is 12 cm3. The volume (in cm3) of a cuboid whose sides are double of the above cuboid is
A. 24
B. 48
C. 72
D. 96
Answer:
Given,
Volume of cuboid =12 cm3
Let dimensions of cuboid = l×b×h
So, l×b×h = 12 …………………….(i)
Dimensions of one another cuboid = 2l ×2b×2h
Volume of cuboid = 8lbh = 8×12 = 96cm3
Question 27.
If the sum of all the edges of acube is 36 cm, then the volume (in cm3) of that cube is
A. 9
B. 27
C. 219
D. 729
Answer:
Given,
Sum of all edges of cube = 36 cm
= 12 a = 36
= a = 
Volume of cube = a3 = 3×3×3 = 27 cm3
Question 28.
The number of cubes of side 3 cm that can be cut from a cuboid of dimensions 10 cm×9 cm×6 cm, is
A. 9
B. 10
C. 18
D. 20
Answer:
Given,
Side of cube = 3cm
Dimension of cuboid = 10 cm×9cm ×6 cm
Volume of cuboid = 10×9×6 = 540 cm3
Volume of cube = a3 = 33 = 27 cm3
Number of cubes can be formed = 
Question 29.
On a particular day, the rain fall recorded in a terrace 6 m long and 5 m board is 15 cm. The quantity of water collected in the terrace is
A. 300 litres
B. 450 litres
C. 3000 litres
D. 4500 litres
Answer:
Given ,
Length of terrace = 6m = 600 cm
Breadth of terrace = 5m = 500 cm
Height of rainfall = 15 cm
Quantity of water collected on roof = 600×500×15 = 450000 cm3
=