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We know that the volume of a cylinder is given by V = πr^2h, where r is the radius and h is the height.
Since the cone and cylinder share the same radius and height, we can calculate the volume of the cone as V_cone = (1/3)Ï€r^2h.
The volume of the cylinder is given as 54 in^3, so we can set up the equation as:
54 = πr^2h
Solving for h, we find:
h = 54 / (Ï€r^2)
Now, we can substitute the value of h into the volume formula of the cone to find the volume of the cone:
V_cone = (1/3)Ï€r^2 * (54 / (Ï€r^2))
V_cone = 54 / 3
V_cone = 18 in^3
Therefore, the volume of the remaining material after carving the cone out of the cylinder is given by:
Volume remaining = Volume of cylinder - Volume of cone
Volume remaining = 54 - 18
Volume remaining = 36 in^3
So the volume of the remaining material is 36 in^3.