A solid hemisphere of radius 10 cm is melted into two cones of equal height. If each cone’s radius is 5 cm, what is the height of each cone?

Correct — A. Melting keeps the volume, so the two cones together hold the hemisphere's volume.

Hemisphere volume: 2⁄3 π × 10³ = 2000π⁄3 cm³
One cone: 1⁄3 π × 5² × h = 25πh⁄3
Two cones: 2 × 25πh⁄3 = 50πh⁄3

Set equal: 50πh⁄3 = 2000π⁄3
Divide by π⁄3: 50h = 2000, so h = 40 cm → option (a)

Why the others are wrong

• (b)
  45 cm — 45 cm makes the cones too big: two cones give 50π × 45⁄3 = 750π cm³, more than the hemisphere's 2000π⁄3 ≈ 666.7π cm³.
• (c)
  50 cm — 50 cm gives two cones of 2500π⁄3 cm³ in total, 500π⁄3 more than the hemisphere holds. Volume cannot grow when metal is melted and recast.
• (d)
  55cm — 55 cm gives 2750π⁄3 cm³ across the two cones, well above the 2000π⁄3 cm³ of metal available from the hemisphere.

Melting and recasting conserves volume. The shape changes but the amount of metal does not, so the original's volume equals the total volume of everything made from it.

The two cones are equal, so each takes half the metal: 1⁄3 π × 25 × h = 1000π⁄3, which again gives h = 40 cm.

Key facts

• Volume of a hemisphere of radius r = 2⁄3 πr³.
• Volume of a cone = 1⁄3 πr²h.
• When a solid is melted and recast, total volume stays the same unless the question states a wastage.

Common traps

• Setting the hemisphere equal to one cone instead of two, which gives h = 80 cm.
• Using 4⁄3 πr³ (full sphere) for the hemisphere, which also doubles the height to 80 cm.

No directly related past PYQ was found.