CAPF AC 2021 Q51: General Mental Ability – Quantitative Aptitude
Eight metallic balls of one centimetre radius each are melted into one ball. The diameter of the new ball is
- (a) 2 cm
- (b) 6 cm
- (c) 4 cm
- (d) 1 cm
Answer: (c) 4 cm
Explanation
Eight small balls melted into one
Given: eight metallic balls of 1 cm radius each are melted into one ball; find the diameter of the new ball.
How to think: melting changes the shape, not the amount of metal.
So the volume stays the same.
Volume of a sphere depends on the cube of the radius.
| Step | Working |
|---|---|
| Volume of one small ball | (4/3) × π × 1³ |
| Volume of eight balls | 8 × (4/3) × π |
| Volume of the big ball | (4/3) × π × R³ |
| Equate | R³ = 8 |
| Radius | R = 2 cm |
| Diameter | 2 × 2 = 4 cm |
Shortcut: 8 times the volume means the radius becomes the cube root of 8, which is 2 times.
Answer: (c) 4 cm.
Why each option is right or wrong
- (a) Incorrect. 2 cm is the radius of the new ball, not its diameter.
- (b) Incorrect. 6 cm would need a radius of 3 cm, which is 27 balls.
- (c) CORRECT. R³ = 8 gives R = 2 cm, so the diameter is 4 cm.
- (d) Incorrect. 1 cm is the radius of each small ball.
Key point
Melting and recasting: volume is conserved. Volume of a sphere = (4/3)πr³. If n equal spheres make one sphere, new radius = r × cube root of n.
Remember
8 equal balls → radius doubles.
Common mistakes
- Stopping at the radius (2 cm) when the question asks for the diameter.
- Multiplying the radius by 8 instead of taking the cube root of 8.
- Adding the radii of the eight balls, as if length and not volume were conserved.
Exam tip
Educated guess: the number 8 is a perfect cube, which tells you the radius simply doubles. Then read the question again: it asks for the diameter.
Sources
- NCERT Class IX Mathematics — Surface Areas and Volumes (volume of a sphere)
- NCERT Class X Mathematics — Surface Areas and Volumes (conversion of solid from one shape to another)