The ratio of the height to the base diameter of an empty ice-cream cone is 3:…
2026
The ratio of the height to the base diameter of an empty ice-cream cone is 3: 2. When a spherical scoop of ice cream is filled in it, such that 1/2 of the scoop is inside the cone, the total length of the ice cream is 36 cm. What is the volume of the ice cream in cc?
- A.
1064π
- B.
972π
- C.
2916π
- D.
243π
Show answer & explanation
Correct answer: B
Concept: When a spherical scoop sits at a cone's rim with half its volume sunk inside the cone and half protruding above the rim, the total visible length of the assembly equals the cone's height plus the sphere's radius — the protruding hemisphere is what adds height beyond the rim. Since the scoop is the only ice cream present, the ice-cream volume equals the volume of the full sphere, (4/3)πr3, not any additional volume from the cone itself.
Let the cone's base radius be r. Since height : diameter = 3 : 2 and diameter = 2r, the cone's height = 3r.
The scoop's sphere shares the cone's rim, so its radius is also r. With half the sphere sunk inside the cone, the extra length showing above the rim equals r.
Total length of the ice cream = cone height + sphere radius = 3r + r = 4r.
Given total length = 36 cm: 4r = 36, so r = 9 cm.
Volume of ice cream = volume of the full sphere = (4/3)π(9)3 = (4/3)π(729) = 972π cc.
Cross-check: 4 × 9 = 36 cm confirms the radius satisfies the total-length condition; and (4/3) × 729 = 972, confirming the volume 972π cc independently.