Use the image to answer the question.
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 m3 . What is the volume of the whole object?(1 point)
To find the volume of the entire object, we need to find the volume of both the hemisphere and the cone and add them together.
Let's start by finding the volume of the cone. The formula for the volume of a cone is V = (1/3)πr^2h, where r is the radius and h is the height.
Given that the volume of the cone is 12 m^3, we can set up the equation as follows:
12 = (1/3)πr^2h
Since we know that the height of the cone is equal to the diameter of the hemisphere, we can let h = 2r.
12 = (1/3)πr^2(2r)
12 = (2/3)πr^3
Now, let's find the volume of the hemisphere. The formula for the volume of a hemisphere is V = (2/3)πr^3.
Since the radius of the hemisphere is the same as the radius of the cone, we can use the same value of r.
So, the volume of the hemisphere is (2/3)πr^3.
To find the volume of the entire object, we add the volume of the cone and the volume of the hemisphere:
Volume of entire object = volume of cone + volume of hemisphere
= (2/3)πr^3 + (2/3)πr^3
= (4/3)πr^3
Therefore, the volume of the whole object is (4/3)πr^3.