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?(1 point)
Responses
36 in.3
36 in. cubed
18 in.3
18 in. cubed
40 in.3
40 in. cubed
27 in.3
27 in. cubed
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The volume of a cone is given by 1/3 * π * r^2 * h, where r is the radius and h is the height.
Since the cone and cylinder have the same radius and height:
Volume of the cone = Volume of the cylinder - Volume of the cone
Volume of the cone = 54 in^3 - 1/3 * π * r^2 * h
Plugging in the values:
Volume of the cone = 54 in^3 - 1/3 * π * r^2 * h
Volume of the cone = 54 in^3 - 1/3 * π * r^2 * h
Volume of the cone = 54 in^3 - 1/3 * π * r^2 * h
Volume of the cone = 54 in^3 - π * (r^2) * (h)/3
Volume of the cone = 54 in^3 - 1/3 * π * (r^2) * h
Since the radius and height are the same:
Volume of the cone = 54 in^3 - 1/3 * π * r^2 * r
Volume of the cone = 54 in^3 - 1/3 * π * r^3
Volume of the cone = 54 in^3 - π * r^3 / 3
Volume of the cone = 54 in^3 - π * (4 in)^3 / 3
Volume of the cone = 54 in^3 - π * 64 in^3 / 3
Volume of the cone = 54 in^3 - 64π in^3 / 3
Volume of the cone ≈ 54 in^3 - 67.07 in^3 ≈ 13.07 in^3
Therefore, the volume of the remaining amount is approximately 13.07 in^3.