A hemisphere is placed on top of an inverted cone. The back of the circumference, the radius, and the perpendicular height of the cone are marked with dashed lines. The radius and the height make a right angle and are perpendicular.
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 call the radius of the hemisphere r and the height of the cone h. Since the height of the cone is equal to the diameter of the hemisphere, we can also say h = 2r.
We know that the volume of a cone is given by the formula: V = (1/3)πr^2h.
We also know that the volume of the cone is 12 m^3. Substituting the known values, we get:
12 = (1/3)πr^2(2r).
Simplifying this equation, we have:
12 = (2/3)πr^3.
Now, we can solve for r:
r^3 = (3/2)(12/π).
r^3 = 18/π.
Taking the cube root of both sides, we get:
r ≈ 2.220 m.
Since the height of the cone is h = 2r, we have:
h ≈ 4.440 m.
Now that we know the radius and height, we can calculate the volume of the hemisphere using the formula: V = (2/3)πr^3.
V = (2/3)π(2.220)^3 ≈ 22.017 m^3.
Finally, to find the volume of the whole object, we add the volumes of the hemisphere and the cone:
Volume of the whole object = Volume of hemisphere + Volume of cone.
Volume of the whole object ≈ 22.017 + 12 ≈ 34.017 m^3.
Therefore, the volume of the whole object is approximately 34.017 cubic meters.