A sculpture created designed by carving a cone out of a cylinder, the cone and cylinder shared 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?
40 in.³
18 in.³
27 in.³
36 in.³
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height. The volume of the cylinder before removing the cone is 54 in^3, so we can set up the equation πr^2h = 54.
Since the cone is carved out of the cylinder, we need to find the volume of the cone. The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height. Since the cone and cylinder have the same height, we can express the height of the cone as h.
Given that the cone and cylinder have the same radius and height, the volume of the cone is (1/3)πr^2h = (1/3)(πr^2)(h/3) = (1/9)πr^2h.
Therefore, the volume remaining after removing the cone is the volume of the cylinder minus the volume of the cone: 54 - (1/9)πr^2h.
Substituting the value of 54 for πr^2h in the equation, we have 54 - (1/9)(54) = 54 - 6 = 48 in^3.
Therefore, the volume of the amount remaining is 48 in^3.
There is no 48
I apologize for the mistake. Let's calculate the correct volume of the amount remaining.
The volume of the cylinder is given as 54 in³, which can be expressed as πr^2h = 54, where r is the radius and h is the height.
Since the cone and cylinder have the same radius and height, the volume of the cone can be expressed as (1/3)πr^2h.
To find the volume remaining after removing the cone, we subtract the volume of the cone from the volume of the cylinder: 54 - (1/3)πr^2h.
Since the cone and cylinder have the same radius and height, their volumes are directly proportional. Thus, the ratio of the volume of the cone to the volume of the cylinder is (1/3).
Multiplying the volume of the cylinder (54 in³) by (1/3) gives us the volume of the cone: (1/3) * 54 = 18 in³.
Therefore, the volume of the amount remaining is 54 - 18 = 36 in³.
Hence, the correct answer is 36 in³.