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?
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height of the cylinder. Since the cone that is carved out shares the same radius and height as the cylinder, its volume can be calculated using the same formula.
The volume of the cylinder before carving the cone is 54 in^3, so we can express this as:
54 = πr^2h
Since the radius and height are the same, we can simplify this equation to:
54 = πr^2(r)
Now, let's solve for r:
54 = πr^3
Dividing both sides of the equation by π and taking the cube root of both sides, we get:
r = (54/π)^(1/3)
Now that we have the value of r, we can calculate the volume of the cone that was carved out. Using the formula V = (1/3)πr^2h, where r is the radius and h is the height (which is the same as the radius of the cylinder), we get:
V = (1/3)π[(54/π)^(1/3)]^2[(54/π)^(1/3)]
Simplifying this equation, we find:
V = (1/3)π[(54/π)^(2/3)][(54/π)^(1/3)]
V = (1/3)π(54/π)^[(2/3)+(1/3)]
V = (1/3)π(54/π)^(3/3)
V = (1/3)π(54/π)
V = (1/3)(54)
V = 18
Therefore, the volume of the remaining amount is 18 in^3.