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?
Let's denote the radius of the hemisphere as r and the height of the cone as 2r (since it is equal to the diameter of the hemisphere).
The volume of the cone is given by the formula V_cone = (1/3)πr^2h, where h is the height of the cone. Substituting the given values, we have:
V_cone = (1/3)πr^2(2r) = 12 m^3
2πr^3/3 = 12
2πr^3 = 36
r^3 = 18/π
r = (18/π)^(1/3)
The volume of the hemisphere is V_hemi = (2/3)πr^3. Therefore, the volume of the whole object is the sum of the volume of the hemisphere and the cone:
V_object = V_cone + V_hemi
V_object = 12 + (2/3)πr^3
V_object = 12 + (2/3)π(18/π)^(1/3)
V_object = 12 + 12^(1/3)
Therefore, the volume of the whole object is approximately 17.55 m^3.