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
27 in.3
36 in.3
40 in.3
18 in.3
The volume of a cone is given by the formula V = (1/3)πr^2h and the volume of a cylinder is given by the formula V = πr^2h.
Since the cone and cylinder share the same radius and height, the volume of the cone that was removed is (1/3)πr^2h.
So, the volume remaining after removing the cone is the volume of the cylinder minus the volume of the cone:
V = πr^2h - (1/3)πr^2h
V = (3/3)πr^2h - (1/3)πr^2h
V = (2/3)πr^2h
Since the original volume of the cylinder was 54 in^3, we have:
(2/3)πr^2h = 54
(2/3)πr^2h = 54
(2/3)*(π*r^2*h) = 54
2*(π*r^2*h)/3 = 54
(π*r^2*h) = 54*3/2
(π*r^2*h) = 81
Therefore, the volume of the amount remaining is 81 in^3.
The closest answer option is 40 in^3.