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.

pls

Answer 1

[AI-generated answer — Bot GPT 3.5]

The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius of the base of the cone and h is the height.

Since the diameter of the ice cream cone is equal to its height, we can let the diameter be d and the radius be r = d/2.

The volume of the ice cream cone is V_cone = (1/3)πr^2h = (1/3)π(d/2)^2(d) = (1/12)πd^3.

The volume of the perfect hemisphere of ice cream on top of the cone is given as 4 in.3.

The formula for the volume of a perfect sphere is V_sphere = (4/3)πr^3, where r is the radius of the sphere.

Since the diameter of the hemisphere is equal to the diameter of the cone, the radius of the hemisphere is r = d/2.

We are given that V_sphere = 4 in.3, so we can solve for d by setting V_sphere = (4/3)πr^3 = (4/3)π(d/2)^3 = (4/3)π(d^3/8) = 4 in.3.

Simplifying, we have (4/3)π(d^3/8) = 4 in.3.
Multiplying both sides by 3/4, we have π(d^3/8) = 3 in.3.
Multiplying both sides by 8/d^3, we have π = (3(8/d^3)) in.3/d^3.
Simplifying, we have π = 24/d^3 in.3/d^3.
Therefore, d^3 = 24/π.
Taking the cube root of both sides, we have d = (24/π)^(1/3).

Now that we have the diameter of the cone, we can find the volume of the cone by plugging it into the formula V_cone = (1/12)πd^3.

V_cone = (1/12)π((24/π)^(1/3))^3 = (1/12)π(24/π) = 2 in.3.

Therefore, the total volume of ice cream in and on top of the cone is 4 in.3 (hemisphere) + 2 in.3 (cone) = 6 in.3.