Question 1. The length, breadth and height of a room are 4 m, 3 m and 3 m respectively. Find the lateral surface area of the room.
Solution: Here, l = 4 m, b = 3 m and h = 3 m.
Lateral surface area = Area of four walls = 2(l + b)h = 2(4 + 3) x 3 m^{2} = 2 x 7 x 3 m^{2 }= 42 m^{2}
∴ The required lateral surface of the room is 42 m^{2}.
Question 2. If the circumference of the base of a right circular cylinder is 110 cm, then find its base area.
Solution: Let r be the radius of the base of the cylinder.
∴ Circumference = 2πr = 2 x(22/7)x r
Now, 2 x(22/7) x r= 110
⇒
Now, Base area =
Question 3. The radii of two cylinders are in the ratio of 2 : 3 and heights are in the ratio of 5 : 3.
Find the ratio of their volumes.
Solution: Ratio of the radii = 2 : 3 Let the radii be 2r and 3r Also, their heights are in the ratio of 5:3 Let the height be 5h and 3h
∴ Ratio of their volumes =
Question 4. If the radius of a sphere is doubled, then find the ratio of their volumes.
Solution: Let the radius of the original sphere = r
∴ Radius of new sphere = 2r
∴ Ratio of their volumes =
Question 5. If the radius of a sphere is such that πr^{2} = 6cm^{2} then find its total surface area.
Solution: ∵ πr^{2} = 6cm^{2 }
∴ Curved S.A. of the hemisphere
= 2 × 6 cm^{2} = 12 cm^{2 }
Also, plane S.A. of the hemisphere = π r^{2} = 6 cm^{2 }
⇒ Total S.A. = C.S.A. + plane S.A. = 12 cm^{2} + 6 cm^{2} = 18 cm^{2}
Question 6. The surface area of a sphere of radius 5 cm is five times the area of the curved surface of a cone of radius 4 cm. Find the height and the volume of the cone (taking π = (22/7).
Solution: Surface area of the sphere = 4πr^{2} = 4 × π × 5 × 5 cm^{2}
Curved surface area of the cone (with slant height as ‘ℓ’) = πrℓ = π × 4 × ℓ cm^{2}
Since,
∴ 4 × π × 5 × 5 = 5 × π × 4 × ℓ
⇒
∴ Volume of the cone =
Question 7. The radius of a sphere is increased by 10%. πrove that the volume will be increased by 33.1% apπroximately.
Solution: The volume of a sphere =
Increased radius =
∴ Increased volume =
= (4/3) x π x 1.331r^{3}
Thus, Increase in volume
∴ Percentage increase in volume
= 0.331 × 100%
= 33.1%.
Question 8. Find the slant height of a cone whose radius is 7 cm and height is 24 cm.
Solution: Here, h = 24 cm and r = 7 cm
Since,
= 25 cm
∴ Slant height = 25 cm.
Question 9. The diameter of a road roller, 120 cm long is 84 cm. If it takes 500 complete revolutions to level a playground, find the cost of levelling it at 2 per square metre.
Solution: Here, radius (r) = 42 cm
Length of the roller = Height of the cylinder
⇒ h = 120 cm
∴ Curved surface area of the roller = 2πrh =2 x(22/7) x 42 x 120 cm^{2 }
= 2 x 22 x 6 x 120 cm^{2} = 31680 cm^{2}
∴ Area levelled in one revolution = 31680 cm^{2}
⇒ Area levelled in 500 revolutions = 31680 x 500 cm^{2}
∴ Cost of levelling the playground = 2 x 15840 = 31680.
Question 10. A conical tent of radius 7 m and height 24 m is to be made. Find the cost of the 5 m wide cloth required at the rate of 50 per metre.
Solution: Radius of the base of the tent (r) = 7 m
Height (h) = 24 m
Slant height (ℓ)=
Now, curved surface area of the conical tent = πrℓ
= (22/7) x 7 x 25 m^{2} = 22 x 25 m^{2} = 550 m^{2 }
Let ‘ℓ’ be the length of the cloth.
∴ ℓ x b = 550
⇒ ℓ x 5 = 550
⇒ ℓ = (550/5) m = 110 m
∴ Cost of the cloth = 50 x 110 = 5500.
Question 11. How many lead balls, each of radius 1 cm, can be made from a sphere of radius 8 cm?
Solution: Radius of the lead ball (r) = 1 cm
∴ Volume of a lead ball = x 1 x 1 x 1 cm^{3}
= (4/3) x (22/7) cm^{3}
Radius of the sphere (r) = 8 cm
∴ Volume of a sphere = x 8 x 8 x 8 cm^{3}
Let the required number of balls = n
∴ [Volume of nlead balls] = [Volume of the sphere]
Thus, the required number of balls is 512.
48 videos378 docs65 tests

1. What is the formula for finding the surface area of a rectangular prism? 
2. How do you find the volume of a cone? 
3. What is the surface area of a sphere? 
4. How do you calculate the total surface area of a cylinder? 
5. How do you find the volume of a pyramid? 
48 videos378 docs65 tests


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