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?
A 40 inches^3
B 27 inches^3
C 36 inches^3
D 18 inches^3
The volume of a cone is given by the formula:
Volume cone = 1/3 * π * r^2 * h,
where r is the radius and h is the height of the cone.
The volume of a cylinder is given by the formula:
Volume cylinder = π * r^2 * h.
Since the cone and cylinder have the same radius and height, we can calculate the volume of the cone that was removed by subtracting the volume of the cone from the volume of the cylinder.
Volume cone = 1/3 * π * r^2 * h = 1/3 * π * (54 in^3) = 18π in^3.
Volume remaining = Volume cylinder - Volume cone = 54 in^3 - 18π in^3 ≈ 54 in^3 - 18 * 3.14159 in^3 ≈ 54 in^3 - 56.54866 in^3 ≈ -2.54866 in^3.
The volume remaining cannot be negative, so we discard this result.
Therefore, the correct answer is not provided among the options given.
If we had to pick one, which is the closest?
The closest option to the calculated result of -2.54866 in^3 would be D) 18 inches^3, but note that this is not the exact answer.
To solve this problem, we need to find the volume of the cone that was carved out of the cylinder, and then subtract it from the volume of the cylinder.
The formula for the volume of a cylinder is Vcylinder = πr^2h, where r is the radius and h is the height.
Given that the volume of the cylinder before carving the cone is 54 in^3, we can write the equation as:
54 = πr^2h
Since the cone that was carved out has the same radius and height as the cylinder, the volume of the cone can be calculated using the formula for the volume of a cone: Vcone = (1/3)πr^2h.
To find the volume of the cone, we can substitute the values of r and h from the cylinder into the formula:
Vcone = (1/3)πr^2h
Now, to find the volume of the remaining object, we subtract the volume of the cone from the original volume of the cylinder:
Volume remaining = Volume of cylinder - Volume of cone
Substituting the respective values, we get:
Volume remaining = 54 in^3 - [(1/3)πr^2h]
Since the radius and height of the cone and cylinder are the same, we can simplify the equation as:
Volume remaining = 54 in^3 - [(1/3) * 54 in^3]
Solving the equation, we get:
Volume remaining = 54 in^3 - 18 in^3 = 36 in^3
Therefore, the volume of the amount remaining is 36 inches^3.
Hence, the correct answer is option C) 36 inches^3.