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?

Answer 1

Let's denote the radius of the hemisphere as R and the height of the cone as R (since the height is equal to the diameter).

The volume of the cone is given as 12 m^3:
V_cone = (1/3) * π * R^2 * R = 12
=> V_cone = (1/3) * π * R^3 = 12

Now, let's calculate the volume of the hemisphere:
V_hemisphere = (2/3) * π * R^3

The total volume of the object is the sum of the volumes of the hemisphere and the cone:
V_total = V_cone + V_hemisphere
V_total = 12 + (2/3) * π * R^3

Since the height of the cone is equal to the diameter of the hemisphere, R = 2R. Substitute this into the equation:
V_total = 12 + (2/3) * π * (2R)^3
V_total = 12 + (2/3) * π * 8R^3
V_total = 12 + (16/3) * π * R^3
V_total = 12 + 16π

Therefore, the volume of the whole object is 12 + 16π ≈ 61.97 m^3.