Question

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

[AI-generated answer — Bot GPT 3.5]

First, let's find the radius of the hemisphere. We know that the volume of a hemisphere is given by the formula:

V = (2/3) * π * r^3
We have V = 4 in^3, so we can solve for r:
4 = (2/3) * π * r^3
Multiply both sides by 3/2:
(3/2) * 4 = π * r^3
12 = π * r^3
Divide both sides by π:
12/π = r^3
Take the cube root of both sides to solve for r:
r ≈ 2.182

The diameter of the ice cream cone is equal to its height, so the radius of the cone is half of the diameter, or 1.091 in.

Now let's find the volume of the cone. Since the height and radius are the same, we can use the formula for the volume of a cone with radius r and height h:
V = (1/3) * π * r^2 * h
V = (1/3) * π * (1.091^2) * 1.091
V ≈ 1.5868 in^3

Finally, let's find the total volume of ice cream in and on top of the cone. This is the volume of the cone plus the volume of the hemisphere:
Total volume = cone volume + hemisphere volume
Total volume ≈ 1.5868 + 4 ≈ 5.5868 in^3

Therefore, the total volume of ice cream in and on top of the cone is approximately 5.5868 cubic inches.