Question

can you help me with this?
Aimee packs ice cream into a cone. She puts a perfect hemisphere of ice cream on top of the cone that has a volume of 4 cubic inches. 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

7th grade level please

Answer 1

Answer: 8 cubic inches.

Work and reasoning (7th-grade level):

1. The hemisphere on top has volume 4 in³. A hemisphere is half a sphere. The volume of a sphere is (4/3)πr³, so the volume of a hemisphere is half that:
hemisphere volume = (1/2)·(4/3)πr³ = (2/3)πr³.

2. The cone’s diameter equals its height. If the cone’s radius is r, then its diameter is 2r, so the height h = 2r.

3. The volume of a cone is (1/3)πr²h. Substitute h = 2r:
cone volume = (1/3)πr²(2r) = (2/3)πr³.

4. Notice that (2/3)πr³ is the same expression as the hemisphere volume. So the cone has the same volume as the hemisphere. Since the hemisphere volume is 4 in³, the cone volume is also 4 in³.

5. Total ice cream = hemisphere + cone = 4 in³ + 4 in³ = 8 in³.

So the total volume of ice cream is 8 cubic inches.