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.
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?
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