Eight metallic balls of one centimetre radius each are melted into one ball. The diameter of the new ball is

Q: What is the answer to CAPF 2021 Paper I Q51?

C. Correct — C, 4 cm. Melting conserves volume, so the volume of the new ball equals the volume of the eight old ones added together. Each small ball has volume (4/3)π(1)³, so the eight together give 8 × (4/3)π = (32/3)π. Setting that equal to (4/3)πR³ leaves R³ = 8 and R = 2 cm. The question asks for the diameter, which is twice the radius, so the answer is 4 cm. The shortcut worth carrying is that volume scales as the cube of a length: eight times the volume is exactly two times every linear dimension, so both the radius and the diameter simply double. (Provisional — pending the official UPSC key.)

(a)2 cm
(b)6 cm
(c)4 cm
(d)1 cm

Answer

Why

Correct — C, 4 cm. Melting conserves volume, so the volume of the new ball equals the volume of the eight old ones added together. Each small ball has volume (4/3)π(1)³, so the eight together give 8 × (4/3)π = (32/3)π. Setting that equal to (4/3)πR³ leaves R³ = 8 and R = 2 cm. The question asks for the diameter, which is twice the radius, so the answer is 4 cm. The shortcut worth carrying is that volume scales as the cube of a length: eight times the volume is exactly two times every linear dimension, so both the radius and the diameter simply double.

Why the others are wrong

(a)2 cm — This is the new radius, not the diameter. Stopping at R = 2 is the single commonest slip on this item, and the examiner has placed the value in the list to catch it.
(b)6 cm — Would need R = 3, so R³ = 27 and a volume 27 times the original. That is what melting twenty-seven balls would give, not eight.
(d)1 cm — The radius of one of the original balls. Combining eight of them cannot leave the size unchanged.

Concept

The volume of a sphere is (4/3)πr³. Because the radius enters as a cube, scaling laws are the fastest route through most sphere questions: multiply the volume by k and every length multiplies by the cube root of k. Melting and recasting problems are built entirely on the fact that volume, and only volume, is preserved — surface area and radius are not.

Notice how little arithmetic is needed once you see the structure. π and the factor 4/3 sit on both sides of the equation and cancel, leaving 8 × 1³ = R³. The exam-hall discipline is to finish the sentence the question actually asked: it says diameter, and three of the four options are lengths that are correct answers to some other question. Reading the last four words of the stem again before ticking is worth more than the algebra here.

Key facts

Volume of a sphere = (4/3)πr³; melting and recasting conserves volume.
Eight balls of radius 1 cm have total volume (32/3)π cubic centimetres.
R³ = 8 gives R = 2 cm, so the diameter is 4 cm.
Volume scales as the cube of length: eight times the volume means twice the radius.
Surface area scales as the square of length, so the recast ball has four times the surface area of one small ball, not eight.

Four lines from eight balls to one

Volume of one small ball→(4/3)π × 1³ = (4/3)π

Total volume of eight→8 × (4/3)π = (32/3)π

Set equal to the new ball→(4/3)πR³ = (32/3)π, so R³ = 8 and R = 2 cm

Answer the question asked→Diameter = 2R = 4 cm

Stop one line early and you tick 2 cm, which is why that value is offered.

Study next

Surface area and volume of the sphere, cylinder and cone, and the relations between themScaling laws — how area and volume respond to a change in lengthRecasting and melting problems where a solid is drawn into a wire or cut into smaller piecesVolume of a hemisphere and of a spherical shellRatio-based mensuration, where the numbers cancel and only the powers matter

Common traps

Reporting the radius when the question asked for the diameter.
Adding radii instead of volumes, which would give 8 cm.
Assuming eight times the volume means eight times the radius.

A one-step mensuration item whose difficulty is entirely in the final word of the stem.

Related PYQs

UPSC_2002_GS1_Q202002Open & attempt →

Consider the volumes of the following: 1. A parallelepiped of length 5 cm, breadth 3 cm and height 4 cm 2. A cube of each side 4 cm 3. A cylinder of radius 3 cm and length 3 cm 4. A sphere of radius 3 cm The volumes of these in the decreasing order is

(a) I, III, II, IV
(b) IV, II, III, I
(c) I, II, III, IV
(d) IV, III, II, I

Answer(d) IV, III, II, I

The same formula used for comparison instead of construction. Four solids are ranked by volume, and the sphere of radius 3 cm at about 113 cubic centimetres comes out on top of a cylinder of the same radius.

CAPF_GAI_2022_Q32022Open & attempt →

A solid spherical ball made of iron is melted and two new balls are made whose diameters are in the ratio of 1 : 2. The ratio of the volume of the smaller new ball to the original ball is

(a) 1 : 3
(b) 1 : 5
(c) 2 : 9
(d) 1 : 9

Answer(d) 1 : 9

The following year's paper ran the same melting rule backwards. There one ball becomes two and the cube law converts a 1 : 2 length ratio into a 1 : 8 volume ratio, so the smaller ball is one part in nine of the original.

Practice

practice — not a real PYQ
Twenty-seven solid spheres of radius 1 cm are melted and recast into one sphere. Its radius is
- (a)3 cm
- (b)9 cm
- (c)27 cm
- (d)6 cm
Answer(a) 3 cm — R³ = 27, so R = 3.
practice — not a real PYQ
If the radius of a sphere is doubled, its volume becomes
- (a)twice
- (b)four times
- (c)six times
- (d)eight times
Answer(d) eight times — volume varies as the cube of the radius.