Question 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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Question 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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Question 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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Question 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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Question 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=8 \ m^3

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Question 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}=17\ m is the longest pole that can be placed in the room

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Question 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= 4.5\ m

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Question 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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Question 9. The surface area of the cuboid is 468 \ cm^2 . Its length and breadth are 12\ cm \ 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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Question 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=3.5\ m

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 

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

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Question 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:

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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Question 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

i) cost of wall paper at a rate of 35Rs/m^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 =4\ m \ 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 

i) Cost of painting the wall =2314\times 35=80990 \ Rs.

ii) cost of carpeting the floor =45\times 30\times 154=207900 \ Rs.

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Question 13 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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Question 14. If the area of the three adjacent faces of a cuboidal box are 120\ cm^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)}=720\ cm^3 

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Question 15. 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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Question 16. 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 100 \ cm^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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Question 17 The external dimensions of a wooden box, open at the top are 54 \ cm \ 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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Question 18. 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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Question 19. 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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Question 20. 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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Question 21. 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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Question 22. 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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Question 23. 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

Question 24. 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

 

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