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Q.1. Find the volume, the total surface area and the lateral surface area of the cuboid having:

  • Length (l)= 24 cm, \ breadth(b)= 16 cm\ and \ height(h) = 7.5 cm
  • Length (l)= 10 m, \ breadth(b)= 35 cm \ and \ height(h) = 1.2 m 

Answer:

a)

Volume of a cuboid =(l\times b\times h)=24\times 16\times 7.5= 2880 cm^3

Total surface Area of a cuboid = 2(lb+bh+lh)  = 2(24\times 16+16\times 7.5+24\times 7.5)  cm^2=1368 cm^2 

Lateral surface Area of a cuboid = 2(l+b)\times h= 2(24+16)\times 7.5 cm^2= 600 cm^2

b)

Volume of a cuboid =(l\times b\times h)=10\times 0.35\times 1.2= 4.2 m^3

Total surface Area of a cuboid =2(lb+bh+lh)  = 2(10\times 0.35+0.35\times 1.2+10\times 1.2)  cm^2=31.84 m^2 

Lateral surface Area of a cuboid  =2(l+b)\times h= 2(10+0.35)\times 1.2 cm^2=24.84 m^2

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Q.2. Find the capacity of a rectangular tub whose length = 6 m , breadth =2.5 m and depth = 1.4 m . Also find the area of the iron sheet required to make the tub.

Answer:

Volume of the tub =(l\times b\times h)=6\times 2.5\times 1.4= 21 m^3

Total surface Area of a cuboid =2(lb+bh+lh)  = 2(6\times 2.5+2.5\times 1.4+1.4\times 6)  m^2=53.8 m^2 

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Q.3. A wall of length 13.5 m , width 60 cm and height 1.6 m is to be constructed by using bricks of dimensions 22.5 cm \ by \ 12 cm \ by\ 8 cm . How many bricks would be needed.

Answer:

Volume of the wall =(l\times b\times h)=13.5\times 0.60\times 1.6= 12.96 m^3

Volume of the brick =(l\times b\times h)=0.225\times 0.12\times 0.08= 0.00216 m^3

Number of bricks needed =  (Volume \ of \ the \ wall)/(Volume \ of \ the \ brick)=12.96/0.00216=6000

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Q.4. How many planks each measuring 5 m \ by\ 24 cm \ by\ 10 cm can be stored in a place 15 m \ long, \ 4 m \ wide \ and\ 60 cm deep?

Answer:

Volume of the place =(l\times b\times h)=15\times 4\times 0.60= 36 m^3

Volume of the plank =(l\times b\times h)=5\times 0.24\times 0.10= 0.12 m^3

Number of planks stored =  (Volume \ of \ the \ place)/(Volume \ of \ the \ plank)=36/0.12=300

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Q.5. A classroom is 10 m \ long, \ 6.4 m \ broad \ and\ 5 m height. If each student is given 1.6 m^2   of the floor area, how many students can be accommodated in the room? How many cubic meters of air would each student get?

Answer:

Area of the floor of the classroom = (l\times b)=10\times 6.4= 64 m^2

Area given to each student = 1.6 m^2

Number of students that can be accommodated in the room =  64/1.6=40 

Cubic meters of air would each student get = 1.6 m^2\times 5m=8m^3

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Q.6. Find the length of the longest pole that can be placed in a room 12 m \ long, 8 m broad and 9 m high.

Answer:

Diagonal of a cuboid =\sqrt{12^2+8^2+9^2}=17m is the longest pole that can be placed in the room

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Q.7. The volume of the cuboid is 972 m^3 . If its length and breadth be 16 m \ and\ 13.5 m respectively, find its height.

Answer:

Volume of a cuboid =(l\times b\times h)

\Rightarrow 972=16\times 13.5\times h

\Rightarrow h= 4.5m

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Q.8. The volume of the cuboid is 1296 m^3 . Its length is 24 m and its breadth and height Are in the ratio of 3:2 . Find the breadth and height of the cube.

Answer:

Volume of a cuboid =(l\times b\times h)

\Rightarrow 1296=24\times 3x\times 2x

\Rightarrow x= 3 m

\Rightarrow Breadth=9 m \ and \ Height \ is \ 6 m

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Q.9. The surface area of the cuboid is 468 cm^2 . Its length and breadth are 12cm \ and\ 9 cm respectively. Find its height.

Answer:

Surface Area of a cuboid = 2(lb+bh+lh)

\Rightarrow 468= 2(12\times 9+9\times h+h\times 12) 

\Rightarrow h=6 m

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Q.10. The length, breadth and height of the room are 8 m, \ 6.5 m \ and\ 3.5 m respectively.  Find: i) the area of the four walls of the room ii)the area of the floor of the room.

Answer:

l=8m, \ b=6.5 m, \ h=3.5m

  1. i) Area of four walls would be =(l\times h+b\times h)\times 2 

=(8\times 3.5+6.5\times 3.5)\times 2=101.5 m^2 

  1. ii) The area of the floor of the room =l\times b=8\times 6.5=52 m^2

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Q.11. A room 9 m \ long, \ 6 m \ wide \ and\ 3.6 m high has one door 1.4 m \ by\ 2 m and two windows each 1.6 m \ by\ 75 cm . Find the: i) area of four walls, excluding the doors and the windows. ii) cost of painting the wall from inside at a rate of 22.50 Rs/m^2 . iii) the cost of painting the ceiling at 25 Rs/m^2 .

Answer:

  1. i) Area of walls excluding the doors are windows

= (l\times h+b\times h)\times 2-(Area \ of \ Doors)\times 1-(Area \ of \ Window)\times 2

=(9\times 3.6+6\times 3.6)\times 2-1.4\times 2-(1.6\times 0.75)\times 2=102.8  m^2 

ii) Cost of painting the wall =102.8\times 22.50=2313 Rs.

iii) Costof painting the ceiling =(9\times 6)\times 25=1350 Rs.

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Q.12. An assembly hall is 45 m \ long, \ 30 m \ broad \ and\ 16 m height. It has five doors, each measuring 4 m \ by\ 3.5 m and four windows 2.5 m \ by \ 1.6 m each. Find the

  1. i) cost of wall paper at a rate of 35Rs/m^2
  2. ii) cost of carpeting the floor at the rate of 154 Rs/m^2 .

Answer:

Wall dimensions: l=45 m, \ b=30 m, \ h=16 m

Door dimensions =4m \ by \ 3.5 m

Window dimensions =2.5 m \ by\ 1.6 m

Area of walls excluding the doors are windows

= (l\times h+b\times h)\times 2-(Area \ of \ Doors)\times 5-(Area \ of \ Window)\times 4

=(45\times 16+30\times 16)\times 2-4\times 3.5\times 5-(2.5\times 1.6)\times 4=2314  m^2 

  1. i) Cost of painting the wall =2314\times 35=80990 Rs.
  2. ii) cost of carpeting the floor =45\times 30\times 154=207900 Rs.

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The length, breadth and height of the cuboid are in the ratio of  7:6:5 . If the surface area of the cuboid is 1926 cm^2 , find its dimensions. Also find the volume of the cuboid.

Answer:

Wall dimensions: l=7x , \ b=6x , \ h=5x

Surface Area of a cuboid = 2(lb+bh+lh) 

\Rightarrow 2(42x^2+30x^2+35x^2 )  cm^2=1926 cm^2 

\Rightarrow x=3

\Rightarrow l=21 cm, \ b=18 cm \ and \ h=15 cm

Volume =21\times 18\times 15=5670 cm^3 

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Q.13. If the area of the three adjacent faces of a cuboidal box are 120cm^2, 72 cm^2  \ and\ 60 cm^2   respectively, then find the volume of the box.

Answer:

Let the  dimensions:  l ,\ b , \ h

l\times b=120

b\times h=72

h\times l=60

Multiplying the above three expressions we get

l^2\times b^2\times h^2=120\times 72\times 60 \Rightarrow Volume= \sqrt{(120\times 72\times 60)}=720cm^3 

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Q.14. A river 2 m deep and 40 m wide is flowing at a rate of 4.5 km/hr . How many cubic meters of water runs into the sea per minute?

Answer:

Rate of flow =(4.5\times 1000)/3600  m/s=1.25 m/s

Volume of water flowing =2\times 40\times 1.25\times 60 =6000  m^3

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Q.15. A closed wooded box 80 cm \ long, \ 65 cm \ wide, \ and\ 45 cm high, is made up of wood 2.5 cm thick. Find i) the capacity of the box, ii) weight of the box if 100cm^3   of wood weighs 8 grams.

Answer:

External Volume of the Box  =(l\times b\times h)=80\times 65\times 45= 234000 cm^3

Internal Length =[80-(2.5+2.5)]=75 cm

Internal Breadth =[65-(2.5+2.5)]=60 cm

Internal Height =[45-(2.5+2.5)]=40 cm

Internal Volume =75\times 60\times 40=  180000 cm^3  

Volume of Wood =234000-180000=4320 gm=4.32 kg

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The external dimensions of a wooden box, open at the top are 54cm \ by \ 30 cm \ by \ 16 cm . It is made up of wood 2 cm thick. Calculate i) the capacity of the box ii) the volume of the wood.

Answer:

External Volume of the Box =(l\times b\times h)

=54\times 30\times 16= 25920 cm^3

Internal Length =[54-(2+2)]=50 cm

Internal Breadth =[30-(2+2)]=26 cm

Internal Height =[16-(2)]=14 cm

Internal Volume =50\times 26\times 14=  18200 cm^3

Volume of Wood =25920-18200=7720 gm=7.72 kg

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Q.16. The internal dimension of the closed box, made up of iron 1 cm thick, are 24 cm by 18 cm by 12 cm . Find the volume of the iron in the box.

Answer:

Internal Volume of the Box =(l\times b\times h)

=24\times 18\times 12= 5184 cm^3

External Length =[24+(1+1)]=26 cm

External Breadth =[18+(1+1)]=20 cm

External Height =[12+(1+1)]=14 cm

External Volume =26\times  20\times  14=7280 cm^3 

Volume of Iron =7280-5184=2096  cm^3  

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Q.17. Find the volume, the total surface area and the lateral surface area and the diagonal of each cube whose edges measures: i) 8 m  ii) 6.5 cm iii) 2 cm 6 mm

Answer:

i)

Volume of a cube = 8^3=512 m^3

Total surface Area of a cube = 6\times 8^2=384  m^2

Lateral surface Area of a cube = 4\times 8^2=256  m^2

Diagonal of a cube = a\sqrt{3}=8\times 1.732=13.856 m

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ii)

Volume of a cube = (6.5)^3=274.625 cm^3

Total surface Area of a cube = 6\times (6.5)^2=253.5  cm^2

Lateral surface Area of a cube = 4\times (6.5)^2=169  cm^2

Diagonal of a cube = a\sqrt{3}=6.5\times 1.732=11.258 cm

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iii)

Volume of a cube = (2.6)^3=17.576 cm^3

Total surface Area of a cube = 6\times (2.6)^2=40.56  cm^2

Lateral surface Area of a cube = 4\times (2.6)^2=27.04  cm^2

Diagonal of a cube = a\sqrt{3}=2.6\times 1.732=4.5032 cm

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Q.18. The surface area of the cube is 1176 cm^2 . Find its volume.

Answer:

Surface Area of a cube = 6a^2=1176

\Rightarrow a=14

Therefore Volume of cube = 14^3=2744 cm^3 

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Q.19. The volume of the cube is 216 cm^3 . Find its surface area.

Answer:

Volume of a cube = a^3=216

\Rightarrow a=6

Surface area =6\times 6^2=216 cm^2

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Q.20. The volume of a cube is 343 cm^3 . Find its surface area.

Answer:

Volume of a cube = a^3=343

\Rightarrow a=7

Surface area =6\times 7^2=294 cm^2

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Q.21. A solid piece of metal in the form of cuboid of dimensions 24 cm \ by\ 18 cm \ by\ 4 cm is melted down and re-casted into a cube. Find the length of each edge of the cube.

Answer:

Volume of a cuboid =(l\times b\times h)=24\times 18\times 4=1728

Let the dimension of cube =a

Volume of Cube = a^3=1728 \Rightarrow 12 cm

Q.22. Three cubes of metal with edges  5 cm \ by\ 4 cm \ by\ 3 cm are melted to form a single cube. Find the lateral surface area of the new cube formed.

Answer:

Let the dimension of the large cube =a

Volume of Large Cube =5^3+4^3+3^3=216= a^3

Therefore the dimension of the large cube =6 cm

Lateral surface Area of a cube = 4\times (6)^2=144  cm^2

ICSE Board: Suggested Books     ICSE Board:  Foundation Mathematics 
Class 8: Reference Books     Class 8: NTSE Preparation 
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