Asked by Fred
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 . 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.
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Answered by
GPT-5 mini
AI
Let r be the radius of the hemisphere (and of the cone's base). The hemisphere has volume 4 in^3, and a full sphere would have twice that volume, 8 in^3. The volume of a sphere is (4/3)πr^3, so
(4/3)πr^3 = 8 => r^3 = 6/π.
The cone's diameter equals its height, so h = 2r. The cone volume is
V_cone = (1/3)πr^2h = (1/3)πr^2(2r) = (2/3)πr^3.
Compare this with the hemisphere volume: hemisphere = (1/2)(4/3)πr^3 = (2/3)πr^3. Thus the cone volume equals the hemisphere volume. Since the hemisphere is 4 in^3, the cone is also 4 in^3.
Total ice cream volume = cone + hemisphere = 4 + 4 = 8 in^3.
(4/3)πr^3 = 8 => r^3 = 6/π.
The cone's diameter equals its height, so h = 2r. The cone volume is
V_cone = (1/3)πr^2h = (1/3)πr^2(2r) = (2/3)πr^3.
Compare this with the hemisphere volume: hemisphere = (1/2)(4/3)πr^3 = (2/3)πr^3. Thus the cone volume equals the hemisphere volume. Since the hemisphere is 4 in^3, the cone is also 4 in^3.
Total ice cream volume = cone + hemisphere = 4 + 4 = 8 in^3.
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