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)
Responses
24 m3
24 m cubed
There is not enough information to say.
There is not enough information to say.
36 m3
36 m cubed
18 m3
To find the volume of the entire object, we need to find the volume of both the hemisphere and the cone and then add them together.
The volume of the cone is given as 12 m^3.
To find the volume of the hemisphere, we need to first find its radius. The radius is equal to half the diameter of the hemisphere, which is equal to the height of the cone.
Since the height of the cone is equal to the diameter of the hemisphere, the radius of the cone is also equal to the radius of the hemisphere.
Let's say the radius of the cone and the hemisphere is r.
The volume of the cone is given by the formula V_cone = (1/3) * π * r^2 * h, where h is the height of the cone.
We know that V_cone = 12 m^3 and h = 2r, so plugging in these values, we get:
12 = (1/3) * π * r^2 * 2r
12 = (2/3) * π * r^3
r^3 = (3/2) * (12/π)
r^3 = 18/π
r ≈ (18/π)^(1/3)
Now, the volume of the hemisphere is given by the formula V_hemisphere = (2/3) * π * r^3.
Plugging in the value of r, we get:
V_hemisphere = (2/3) * π * ((18/π)^(1/3))^3
V_hemisphere = (2/3) * π * (18/π)
V_hemisphere = (2/3) * 18
V_hemisphere = 12 m^3
So the volume of the entire object, which is the sum of the volume of the cone and the hemisphere, is:
V_object = V_cone + V_hemisphere
V_object = 12 + 12
V_object = 24 m^3
The volume of the whole object is 24 m^3.