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 m³ what is the volume volume of the whole object.
Let's denote the radius of the hemisphere as r.
Given that the height of the cone is equal to the diameter of the hemisphere, the height of the cone (h) is equal to 2r.
The volume of a cone is given by the formula V_cone = (1/3) * π * r^2 * h, where r is the radius and h is the height of the cone.
Therefore, we have V_cone = (1/3) * π * r^2 * 2r
Given that the volume of the cone is 12 m³, we can set up the equation as follows:
12 = (1/3) * π * r^2 * 2r
Simplifying, we have:
12 = (2/3) * π * r^3
Multiplying both sides by 3/(2π), we get:
(3/(2π)) * 12 = r^3
36/(2π) = r^3
18/π = r^3
Taking the cube root of both sides, we get:
r = (18/π)^(1/3)
Now, let's find the volume of the hemisphere.
The volume of a hemisphere is given by the formula V_hemisphere = (2/3) * π * r^3
Substituting the value of r, we have:
V_hemisphere = (2/3) * π * ((18/π)^(1/3))^3
Simplifying, we get:
V_hemisphere = (2/3) * π * 18
Now, let's find the volume of the whole object, which is the sum of the volume of the hemisphere and the cone:
V_total = V_cone + V_hemisphere
V_total = (1/3) * π * r^2 * 2r + (2/3) * π * 18
Substituting the values of r and the volume of the cone (V_cone = 12 m³), we have:
V_total = (1/3) * π * ((18/π)^(1/3))^2 * 2 * ((18/π)^(1/3)) + (2/3) * π * 18
Simplifying, we get:
V_total = 2 * ((18/π)^(1/3))^3 + 12
Therefore, the volume of the whole object is 2 * ((18/π)^(1/3))^3 + 12 m³.