Mensuration
- A large solid sphere is melted and moulded to form identical right circular cones with base radius and height same as the radius of the sphere. One of these cones is melted and moulded to form a smaller solid sphere. Then the ratio of the surface area of the smaller to the surface area of the larger sphere is
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Radius of larger sphere = R units
∴ Its volume = 4 πR³ cu. units 3 Volume of smaller cone = 1 πR³ cu. units 3 Volume of smaller sphere = 1 πR³ 3 ∴ 4 πr³ = 1 πR³ 3 3 ⇒ r³ = R³ 4 ⇒ r = R ³√4
∴ Surface area of smaller sphere : Surface area of larger sphere
= 4πr² : 4πR²
= r² : R²= 
R 
² : R² = 1 : (³√4)² ³√4
= 1 : [{(2²)}¹/3]³ = 1 : 24/3Correct Option: D
Radius of larger sphere = R units
∴ Its volume = 4 πR³ cu. units 3 Volume of smaller cone = 1 πR³ cu. units 3 Volume of smaller sphere = 1 πR³ 3 ∴ 4 πr³ = 1 πR³ 3 3 ⇒ r³ = R³ 4 ⇒ r = R ³√4
∴ Surface area of smaller sphere : Surface area of larger sphere
= 4πr² : 4πR²
= r² : R²= R ² : R² = 1 : (³√4)² ³√4
= 1 : [{(2²)}¹/3]³ = 1 : 24/3
- A wooden box of dimensions 8 metre × 7 metre × 6 metre is to carry rectangular boxes of dimensions 8 cm × 7 cm × 6 cm. The maximum number of boxes that can be carried in 1 wooden box is
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Volume of wooden box = (8 × 7 × 6) cu.m.
= (8 × 7 × 6 ×1003) cu.cm.
Volume of a box = (8 × 7× 6) cu.cm.∴ Maximum number of boxes = 8 × 7 × 6 × 100³ = 1000000 8 × 7 × 6 Correct Option: D
Volume of wooden box = (8 × 7 × 6) cu.m.
= (8 × 7 × 6 ×1003) cu.cm.
Volume of a box = (8 × 7× 6) cu.cm.∴ Maximum number of boxes = 8 × 7 × 6 × 100³ = 1000000 8 × 7 × 6
- Two circular cylinders of equal volume have their heights in the ratio 1 : 2. Ratio of their radii is (Take π = 22/7)
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πr1²h1 = πr2²h2
⇒ r1 ² = h2 = 2 r2 h1 1 ∴ r1 = √2 = √2 : 1 r2 2 Correct Option: C
πr1²h1 = πr2²h2
⇒ r1 ² = h2 = 2 r2 h1 1 ∴ r1 = √2 = √2 : 1 r2 2
- A rectangular piece of paper of dimensions 22 cm by 12 cm is rolled along its length to form a cylinder. The volume (in cu.cm.) of the cylinder so formed is ( use π = 22/7 )
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Paper is folded along the length.
∴ Circumference of the base = 22 cm,
Height of cylinder = 12 cm
∴ 2π r = 22⇒ 2 × 22 × r = 22 7
⇒ r = 7/2 cm
∴ Volume of cylinder = πr²h= 22 × 7 × 7 × 12 = 462 cu.cm. 7 2 2 Correct Option: C
Paper is folded along the length.
∴ Circumference of the base = 22 cm,
Height of cylinder = 12 cm
∴ 2π r = 22⇒ 2 × 22 × r = 22 7
⇒ r = 7/2 cm
∴ Volume of cylinder = πr²h= 22 × 7 × 7 × 12 = 462 cu.cm. 7 2 2
- A sphere is placed inside a right circular cylinder so as to touch the top, base and the lateral surface of the cylinder. If the radius of the sphere is R, the volume of the cylinder is
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Height of cylinder = 2 R units
Radius of base = R units
∴ Volume of cylinder = πR²h
⇒ π(R)²(2R) = 2πR³Correct Option: A
Height of cylinder = 2 R units
Radius of base = R units
∴ Volume of cylinder = πR²h
⇒ π(R)²(2R) = 2πR³
