Question

A hemisphere is placed on top of an inverted cone. The back of the circumference, the radius, and the perpendicular height of the cone are marked with dashed lines. The radius and the height make a right angle and are perpendicular.
Aimee packs ice cream into an ice cream cone. She then puts a perfect hemisphere of ice cream on top of the cone that has a volume of 4 in.3
The diameter of the ice cream cone is equal to its height. What is the total volume of ice cream in and on top of the cone? Use the relationship between the formulas for the volumes of cones and spheres to help solve this problem. Show your work and explain your reasoning.

Answer 1

Let the common radius of the hemisphere and the cone opening be r.

Hemisphere volume:
V_hem = (1/2)·(4/3)πr^3 = (2/3)πr^3 = 4 in^3.

Cone volume (radius r, height = diameter = 2r):
V_cone = (1/3)πr^2h = (1/3)πr^2(2r) = (2/3)πr^3.

But (2/3)πr^3 = V_hem = 4 in^3, so V_cone = 4 in^3.

Total ice cream volume = V_hem + V_cone = 4 + 4 = 8 in^3.

(For reference, r^3 = 6/π so r ≈ 1.24 in.)