Use the image to answer the question.
A 3-dimensional cone is placed on top of a 3-dimensional cylinder. One of the cylinder bases is common to the cone, with the same circumference.
A 3D modeler is designing a component of a castle in a video game. She places a cone on top of a cylinder so they line up and share the same radius. The cone and the cylinder share the same height. If the volume of the cone is 50 cm3
, what is the volume of the whole object?
To find the volume of the whole object, we need to find the volume of the cylinder and then add it to the volume of the cone.
The volume of a cone is given by the formula:
Vcone = (1/3) * π * r^2 * h
where r is the radius of the cone and h is the height of the cone.
Since the cone and the cylinder share the same radius and height, the volume of the cylinder will be:
Vcylinder = π * r^2 * h
Given that the volume of the cone is 50 cm^3, we have:
50 = (1/3) * π * r^2 * h
We also know that the volume of the whole object is the sum of the volumes of the cone and the cylinder:
Vwhole = Vcone + Vcylinder
To find the volume of the whole object, we need to find the value of Vcylinder.
From the equation 50 = (1/3) * π * r^2 * h, we can rearrange it to solve for r^2:
r^2 = (50 * 3) / (π * h)
Substituting this value of r^2 into the equation for Vcylinder, we get:
Vcylinder = π * ((50 * 3) / (π * h)) * h
Simplifying, we have:
Vcylinder = 150 cm^3
Now we can find the volume of the whole object:
Vwhole = Vcone + Vcylinder
Vwhole = 50 cm^3 + 150 cm^3
Vwhole = 200 cm^3
Therefore, the volume of the whole object is 200 cm^3.