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

i)  $Length (l)= 24 \ cm, \ breadth(b)= 16 \ cm\ and \ height(h) = 7.5 \ cm$

ii) $Length (l)= 10 \ m, \ breadth(b)= 35 \ cm \ and \ height(h) = 1.2 \ m$

i) 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 $= 2(l+b)\times h= 2(24+16)\times 7.5 cm^2= 600 \ cm^2$

ii) 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.

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.

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?

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?

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.

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.

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.

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.

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.

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

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 $25 Rs/m^2$.

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 $154 Rs/m^2$.

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.

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.

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?

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.

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.

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.

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) $2 \ cm 6 \ mm$

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 $= a\sqrt{3}=8\times 1.732=13.856 \ m$

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 $= a\sqrt{3}=6.5\times 1.732=11.258 \ cm$

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.

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.

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.

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.

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.

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 $= 4\times (6)^2=144 \ cm^2$