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
40 in.3
40 in. cubed
18 in.3
18 in. cubed
27 in.3
choose one of the answers
The volume of a cylinder is given by the formula V = πr²h, where r is the radius and h is the height.
Since the cone and cylinder share the same radius and height, the volume of the cone can be found by subtracting the volume of the cylinder from the total volume.
The volume of the cylinder before removing the cone is given as 54 in.3.
Let's plug in the values into the formula and solve for the radius.
54 = πr²h
54 = πr²r (since h = r)
54 = πr³
Divide both sides of the equation by π to isolate r³:
r³ = 54/π
Take the cube root of both sides to solve for r:
r = (54/π)^(1/3)
Now, let's calculate the volume of the cone using the formula V = (1/3)πr²h:
V of cone = (1/3)πr²h
V of cone = (1/3)π(54/π)²(54/π) (since r = (54/π)^(1/3) and h = r)
V of cone = (1/3)(54/π)(54/π)(54/π)
V of cone = 54²(54/π³)
V of cone = 2916/π²
Now, let's find the volume of the remaining amount by subtracting the volume of the cone from the volume of the cylinder:
Remaining volume = Volume of cylinder - Volume of cone
Remaining volume = 54 - 2916/π²
Remaining volume ≈ 54 - 29.321
Remaining volume ≈ 24.679
Therefore, the volume of the amount remaining is approximately 24.679 in.3.