An engineer wants to know the volume of the following object composed of a hemisphere and a cone she knows that the height of the cone is equal to the diameter of the hemisphere and the volume of the cone is 12 m cubed what is the volume of the whole entire
To find the volume of the entire object, we need to calculate the volumes of both the hemisphere and the cone, and then add them together.
Let's start with the volume of the cone. We know that the volume of a cone is given by the formula:
V_cone = (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 can express h in terms of the radius r as
h = 2r.
Substituting this value of h in the volume formula, we have
12 = (1/3)πr^2(2r),
12 = (2/3)πr^3.
Thus, we can solve for r:
r^3 = (3/2)(12/π),
r^3 = 18/π,
r = (18/π)^(1/3).
Now, let's find the volume of the hemisphere. The formula for the volume of a hemisphere is:
V_hemisphere = (2/3)πr^3.
Substituting the value of r we found earlier, we have
V_hemisphere = (2/3)π[(18/π)^(1/3)]^3,
V_hemisphere = (2/3)π(18/π),
V_hemisphere = 12π.
Finally, the volume of the entire object is the sum of the volume of the hemisphere and the volume of the cone:
V_total = V_hemisphere + V_cone,
V_total = 12π + 12,
V_total ≈ 37.7 m^3.
Therefore, the volume of the whole object is approximately 37.7 cubic meters.