Use the image to answer the question.
A cone is placed inside a cylinder. The apex of the cone touching the center of the top circle of the cylinder is highlighted with a dot. The cone with its base is drawn in dashed lines. The base of the cone is common with the base of the cylinder.
A sculptor is creating a design by carving a cone out of a cylinder. The cone and cylinder share the same radius and height. If the volume remaining is 144 in.3
, what was the volume of the original cylinder?
To find the volume of the original cylinder, we need to subtract the volume of the cone that was carved out from the total volume of the cylinder.
Since the cone and cylinder have the same radius and height, we can use the formula for the volume of a cone to find the volume of the carved out cone.
The formula for the volume of a cone is: Vcone = (1/3)πr²h, where r is the radius and h is the height.
Since the cone and the cylinder share the same radius and height, we can write the volume of the cone as Vcone = (1/3)πr²h.
Given that the volume remaining is 144 in³, we can set up the equation:
144 = Vcylinder - Vcone
Since the cone has been carved out of the cylinder, the volume remaining is equal to the volume of the original cylinder minus the volume of the carved out cone.
Substituting the formula for the volume of the cone, we have:
144 = Vcylinder - (1/3)πr²h
We can solve this equation for the volume of the original cylinder, Vcylinder:
Vcylinder = 144 + (1/3)πr²h
Therefore, the volume of the original cylinder is 144 + (1/3)πr²h.