Mensuration
- The volume of a right circular cone which is obtained from a wooden cube of edge 4.2 dm wasting minimum amount of word is :
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The volume of cone should be maximum.∴ Radius of the base of cone = Edge of cube 2 = 4.2 = 2.1 dm. 2
Height = Edge of cube = 4.2 dm.∴ Volume of cone = 1 πr²h 3 = 
1 × 22 × 2.1 × 2.1 × 4.2 
cu.cm. 3 7
= 19.404 cu. dm.Correct Option: C
The volume of cone should be maximum.∴ Radius of the base of cone = Edge of cube 2 = 4.2 = 2.1 dm. 2
Height = Edge of cube = 4.2 dm.∴ Volume of cone = 1 πr²h 3 = 1 × 22 × 2.1 × 2.1 × 4.2 cu.cm. 3 7
= 19.404 cu. dm.
- Base of a right prism is a rectangle, the ratio of whose length and breadth is 3 : 2. If the height of the prism is 12 cm and total surface area is 288 sq. cm., the volume of the prism is :
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Length of base = 3x cm and breadth = 2x cm (let)
Total surface area of prism = perimeter of base × height + 2 × area of base
= [2(3x + 2x) × 12 + 2 × 3x × 2x] sq. cm.
= (120x + 12x2) sq. cm.
According to the question,
120x + 12x² = 288
⇒ x² + 10x= 24
⇒ x² + 10x – 24 = 0
⇒ x² + 12x – 2x – 24 = 0
⇒ x (x + 12) – 2(x + 12) = 0
⇒ (x – 2) (x +12) = 0
⇒ x = 2 because x ≠ –12
∴ Volume of prism = Area of base × height = (3x × 2x × 12) cu. cm.
= 72x² = (72 × 2 × 2) cu. cm.
= 288 cu. cm.Correct Option: B
Length of base = 3x cm and breadth = 2x cm (let)
Total surface area of prism = perimeter of base × height + 2 × area of base
= [2(3x + 2x) × 12 + 2 × 3x × 2x] sq. cm.
= (120x + 12x2) sq. cm.
According to the question,
120x + 12x² = 288
⇒ x² + 10x= 24
⇒ x² + 10x – 24 = 0
⇒ x² + 12x – 2x – 24 = 0
⇒ x (x + 12) – 2(x + 12) = 0
⇒ (x – 2) (x +12) = 0
⇒ x = 2 because x ≠ –12
∴ Volume of prism = Area of base × height = (3x × 2x × 12) cu. cm.
= 72x² = (72 × 2 × 2) cu. cm.
= 288 cu. cm.
- A right prism has a triangular base whose sides are 13 cm, 20 cm and 21 cm. If the altitude of the prism is 9 cm, then its volume is
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In ∆ABC, a = 13 cm., b = 20 cm., c = 21 cm.,
Semi-perimeter = s = a + b + c = 13 + 20 + 21 cm. 2 2 = 54 = 27 cm. 2
∴ Area of ∆ABC =
∴ Area of the base of prism = √s(s - a)(s - b)(s - c)
= √27(27 - 13)(27 - 20)(27 - 21) = √27 × 14 × 7 × 6
= √3 × 3 × 3 × 2 × 7 × 7 × 2 × 3
= 3 × 3 × 2 × 7 = 1236 sq.cm.
∴ Volume of prism = Area of base × height
= 126 × 9 = 1134 cu.cm.Correct Option: B
In ∆ABC, a = 13 cm., b = 20 cm., c = 21 cm.,
Semi-perimeter = s = a + b + c = 13 + 20 + 21 cm. 2 2 = 54 = 27 cm. 2
∴ Area of ∆ABC =
∴ Area of the base of prism = √s(s - a)(s - b)(s - c)
= √27(27 - 13)(27 - 20)(27 - 21) = √27 × 14 × 7 × 6
= √3 × 3 × 3 × 2 × 7 × 7 × 2 × 3
= 3 × 3 × 2 × 7 = 1236 sq.cm.
∴ Volume of prism = Area of base × height
= 126 × 9 = 1134 cu.cm.
- Volume of a right circular cylinder of height 21 cm and base radius 5 cm is :
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Volume of right circular cylinder = πr²h
= 22 × 5 × 5 × 21 = 1650 cu. cm. 7 Correct Option: D
Volume of right circular cylinder = πr²h
= 22 × 5 × 5 × 21 = 1650 cu. cm. 7
- The volume of the largest right circular cone that can be cut out of a cube of edge 7 cm ? (Use π = 22/7)
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Radius of the base of cone = 7 cm. 2
Its height = 7 cm∴ Volume of cone = 1 πr²h 3 = 1 × 22 × 7 × 7 × 7 cu.cm. 3 7 2 2 = 539 = 89.83 cu.cm. 6 Correct Option: B
Radius of the base of cone = 7 cm. 2
Its height = 7 cm∴ Volume of cone = 1 πr²h 3 = 1 × 22 × 7 × 7 × 7 cu.cm. 3 7 2 2 = 539 = 89.83 cu.cm. 6
