Mensuration
- A spherical lead ball of radius 6 cm is melted and small lead balls of radius 3 mm are made. The total number of possible small lead balls is :
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Volume of larger ball = 4 π × (6)³ cu.cm. 3 Volume of a smaller ball = 4 π 
3 
³ cu.cm. 3 10 ∴ Number of smaller balls = 4 π × 6 × 6 × 6 3 4 π × 3 × 3 × 3 10 3 10 10 = 6 ×6 ×6 ×1000 = 8000 3 ×3 ×3 Correct Option: D
Volume of larger ball = 4 π × (6)³ cu.cm. 3 Volume of a smaller ball = 4 π 3 ³ cu.cm. 3 10 ∴ Number of smaller balls = 4 π × 6 × 6 × 6 3 4 π × 3 × 3 × 3 10 3 10 10 = 6 ×6 ×6 ×1000 = 8000 3 ×3 ×3
- The heights of a cone and a cylinder are equal. The radii of their bases are in the ratio 2 : 1. The ratio of their volumes is :
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Radius of the base of cone = 2r units
Radius of the base of cylinder = r units
Height of cone = height of cylinder = h units∴ Required ratio = 1 π(2r)² × h 3 πr²h = 4 πr²h = 4 : 4 : 3 3 πr²h 3 Correct Option: A
Radius of the base of cone = 2r units
Radius of the base of cylinder = r units
Height of cone = height of cylinder = h units∴ Required ratio = 1 π(2r)² × h 3 πr²h = 4 πr²h = 4 : 4 : 3 3 πr²h 3
- The external and the internal radii of a hollow right circular cylinder of height 15 cm. are 6.75 cm. and 5.25 cm. respectively. If it is melted to form a solid cylinder of height half of the original cylinder, then the radius of the solid cylinder is
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Volume of the metal of hollow cylinder = π(R² – r²) h
= π (6.752 – 5.252) × 15
= π (6.75 + 5.25) (6.75 – 5.25) × 15
= π × 12 × 1.5 × 15 cu. cm.
If the radius of the base of solid cylinder be r1 cm, then
πr1² h1 = π × 12 × 1.5 × 15⇒ r1² × 15 = 12 × 1.5 × 15 2
= 12 × 1.5 × 2
⇒ r1² = 36
⇒ r1 = 36 = 6 cm.Correct Option: A
Volume of the metal of hollow cylinder = π(R² – r²) h
= π (6.752 – 5.252) × 15
= π (6.75 + 5.25) (6.75 – 5.25) × 15
= π × 12 × 1.5 × 15 cu. cm.
If the radius of the base of solid cylinder be r1 cm, then
πr1² h1 = π × 12 × 1.5 × 15⇒ r1² × 15 = 12 × 1.5 × 15 2
= 12 × 1.5 × 2
⇒ r1² = 36
⇒ r1 = 36 = 6 cm.
- The radius of a sphere and right circular cylinder is ‘r’ units. Their volumes are equal. The ratio of the height and radius of the cylinder is :
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Volume of sphere = Volume of cylinder
⇒ 4 πr³ = πr²h 3
⇒ 4r = 3h⇒ h = 4 = 4 : 3 r 3 Correct Option: D
Volume of sphere = Volume of cylinder
⇒ 4 πr³ = πr²h 3
⇒ 4r = 3h⇒ h = 4 = 4 : 3 r 3
- The radius of cross section of a solid right circular cylindrical rod is 3.2 dm. The rod is melted and 44 equal solid cubes of side 8 cm are formed. The length of the rod is : (Take π = 22/7)
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Volume of cylindrical rod = 44 × volume of solid cube
⇒ πr²h = 44 × (edge)³⇒ 22 × 32 × 32 × h 7
= 44 × 8 × 8 × 8⇒ h = 44 × 8 × 8 × 8 × 7 = 7 cm. 22 × 32 × 32 Correct Option: B
Volume of cylindrical rod = 44 × volume of solid cube
⇒ πr²h = 44 × (edge)³⇒ 22 × 32 × 32 × h 7
= 44 × 8 × 8 × 8⇒ h = 44 × 8 × 8 × 8 × 7 = 7 cm. 22 × 32 × 32
