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)
If the cone and the cylinder have the same radius and height, and the cone is carved out of the cylinder, then the volume of the remaining shape would be the volume of the cylinder minus the volume of the cone.
The volume of a cylinder is given by the formula V_cylinder = πr^2h, where r is the radius and h is the height.
The volume of a cone is given by the formula V_cone = (1/3)πr^2h, where r is the radius and h is the height.
Since the cone and the cylinder have the same radius and height, we can substitute the same values for r and h in both formulas.
Let's say the radius and height of the cone and cylinder are both equal to x.
Therefore, the volume of the cylinder before removing the cone is V_cylinder = πx^2x = πx^3.
And the volume of the cone is V_cone = (1/3)πx^2x = (1/3)πx^3.
The volume of the remaining shape is V_remaining = V_cylinder - V_cone.
Substituting the values, we get:
V_remaining = πx^3 - (1/3)πx^3.
Simplifying, we get:
V_remaining = (2/3)πx^3.
We are given that the volume of the cylinder before removing the cone is 54 in^3.
Therefore, 54 = πx^3.
To find the volume of the remaining amount, we need to solve for x and substitute it into the formula V_remaining = (2/3)πx^3.
Dividing both sides of the equation by π, we get:
54/π = x^3.
Taking the cube root of both sides, we get:
x = (54/π)^(1/3).
Substituting this value into the formula for V_remaining, we get:
V_remaining = (2/3)π((54/π)^(1/3))^3.
Simplifying, we get:
V_remaining = (2/3)π * 54/π.
Canceling out π, we get:
V_remaining = (2/3) * 54.
Finally, calculating the value, we get:
V_remaining = 36 cubic inches.
Therefore, the volume of the remaining amount is 36 in^3.