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. (4 points)

Answer 1

[AI-generated answer — Bot GPT 3.5]

Let's call the height and the diameter of the ice cream cone "x". Since the diameter is equal to the height, the radius of the cone is also "x/2".

The volume of a cone is given by the formula Vcone = (1/3) * π * r^2 * h.

In this case, Vcone = (1/3) * π * (x/2)^2 * x = (1/3) * π * x^3/4.

The volume of a hemisphere is given by the formula Vhemisphere = (2/3) * π * r^3.

In this case, Vhemisphere = (2/3) * π * (x/2)^3 = (2/3) * π * x^3/8.

The total volume of ice cream is the sum of the volumes of the cone and the hemisphere: Vtotal = Vcone + Vhemisphere.

Substituting the formulas for Vcone and Vhemisphere, we have:
Vtotal = (1/3) * π * x^3/4 + (2/3) * π * x^3/8.

By finding a common denominator, we can simplify this expression:
Vtotal = (2/6) * π * x^3/8 + (3/6) * π * x^3/8 = (5/6) * π * x^3/8.

The total volume of ice cream in and on top of the cone is (5/6) * π * x^3/8.