A cylinder and hemisphere have the same radius. Their combined height is 30 cm. If the cylinder and hemisphere have equal volumes, find the radius.

Correct — D. Let the common radius be r and the cylinder's height h. A hemisphere is r tall.

Equal volumes: πr²h = (2⁄3)πr³
Cancel πr²: h = (2⁄3)r

Combined height: h + r = 30
(2⁄3)r + r = (5⁄3)r = 30

r = 30 × 3⁄5 = 18 cm → option (d)

Why the others are wrong

• (a)
  12cm — 12 cm is the cylinder's height, h = 30 − 18. The question asks for the radius, the larger part of the 30 cm.
• (b)
  15cm — 15 cm splits the height evenly, h = r = 15. Then the cylinder holds 3375π and the hemisphere 2250π, so equal heights do not give equal volumes.
• (c)
  16cm — 16 cm leaves h = 14, but equal volumes need h = (2⁄3) × 16 ≈ 10.67 cm. The cylinder would hold 3584π against the hemisphere's ≈ 2730.7π.

Equal volumes fix the shape. A cylinder πr²h equals a hemisphere (2⁄3)πr³ of the same radius exactly when h = (2⁄3)r, so the cylinder is shorter than the hemisphere is tall.

Combined height means the hemisphere stands on the cylinder: total = h + r. Heights in the ratio h : r = 2 : 3 split 30 cm into 12 and 18.

Key facts

• Volume of a cylinder = πr²h, of a hemisphere = (2⁄3)πr³.
• Equal volume at equal radius means h = (2⁄3)r.
• Check: π × 18² × 12 = 3888π and (2⁄3)π × 18³ = 3888π.

Common traps

• Giving h = 12 cm, the cylinder's height, as the answer.
• Using (4⁄3)πr³, the full sphere's volume, for the hemisphere: that gives h = (4⁄3)r and r ≈ 12.86 cm, not an option.

No directly related past PYQ was found.