Asked by AxAxSU
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 6 inches cubed. the diameter of the ice cream is equal to its height. what is the total volume of ice cream in and top of the cone? use the relationship between volume the formulas for volumes of cone and spheres to help solve this problem. show your and explain your reasoning.
All Answers 1
Answered by
GPT-5 mini
AI
Let r be the radius of the hemisphere (so its diameter is 2r). The hemisphere volume is
V_hemi = (1/2)*(4/3)π r^3 = (2/3)π r^3 = 6 in^3.
The cone’s diameter equals the hemisphere’s diameter, and the problem says the diameter equals the cone’s height, so the cone’s height h = 2r and its base radius = r. The cone volume is
V_cone = (1/3)π r^2 h = (1/3)π r^2 (2r) = (2/3)π r^3.
Thus V_cone = (2/3)π r^3 = V_hemi = 6 in^3.
Total ice cream volume = V_hemi + V_cone = 6 + 6 = 12 in^3.
V_hemi = (1/2)*(4/3)π r^3 = (2/3)π r^3 = 6 in^3.
The cone’s diameter equals the hemisphere’s diameter, and the problem says the diameter equals the cone’s height, so the cone’s height h = 2r and its base radius = r. The cone volume is
V_cone = (1/3)π r^2 h = (1/3)π r^2 (2r) = (2/3)π r^3.
Thus V_cone = (2/3)π r^3 = V_hemi = 6 in^3.
Total ice cream volume = V_hemi + V_cone = 6 + 6 = 12 in^3.
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