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 created a design by carving a cone out of a cylinder. The cone and cylinder share the same radius and height. If the volume of the cylinder before removing the cone is 54 in.^3 , what is the volume of the amount remaining?
Since the cone and cylinder have the same radius and height, they are similar in shape. Therefore, the volume of the cone is equal to one-third of the volume of the cylinder.
Let V be the volume of the cone and cylinder before removing the cone. We know that V is equal to 54 in.^3.
The volume of the cone is given by the formula Vcone = (1/3)πr^2h, where r is the radius and h is the height of the cone.
Since the cone and cylinder have the same radius and height, we can let r and h represent the radius and height of both the cone and the cylinder.
Therefore, Vcone = (1/3)πr^2h = (1/3)πr^2h = (1/3)πr^2h.
Substituting the known values, we have V = (1/3)πr^2h = (1/3)π(3^2)(3).
Simplifying, V = 9π in.^3.
Therefore, the volume of the remaining amount after removing the cone is Vremaining = V - Vcone = 54 - 9π = 54 - 9(3.14159) = 54 - 28.2743 ≈ 25.7257 in.^3.