Question

An engineer wants to know the volume of the following object composed of a hemisphere and a cone.

She knows the height of the cone is equal to the diameter of the hemisphere, and the volume of the cone is 12 m^3. What is the
volume of the whole object?
A.) 36 m^3
B.) 24 m^3
C.) 18 m^3
D.) There is not enough information to say.

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 of the cone.

Given that the volume of the cone is 12 m^3, and the height of the cone is equal to the diameter of the hemisphere, we know that the radius of the base of the cone is equal to the radius of the hemisphere.

Let's denote the radius of the hemisphere and cone as r, and the height of the cone (and diameter of the hemisphere) as 2r.

Since the volume of the cone is 12 m^3:
V_cone = (1/3)πr^2(2r) = 12
2πr^3 = 36
r^3 = 18
r = ∛18 ≈ 2.62 m

The volume of a hemisphere is given by the formula V = (2/3)πr^3. Therefore, the volume of the hemisphere is:
V_hemisphere = (2/3)π(2.62)^3 ≈ 15.38 m^3

The total volume of the object (hemisphere + cone) is the sum of the volumes of the hemisphere and cone:
V_total = V_hemisphere + V_cone ≈ 15.38 + 12 = 27.38 m^3

Therefore, the closest option is:
B.) 24 m^3