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 27 in.3 27 in. cubed 18 in.3 18 in. cubed 36 in.3 36 in. cubed 40 in.3
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height. Since the cone is carved out of the cylinder, the volume of the remaining shape would be the volume of the cylinder minus the volume of the cone.
Given that the volume of the cylinder is 54 in^3, and the cone and cylinder share the same radius and height, the radius and height of the cylinder would be the same for the cone as well.
So, the volume of the cylinder is V = πr^2h = 54 in^3.
Let's call the radius and height of the cylinder (and the cone) as r and h respectively.
Since both the cylinder and the cone share the same radius and height, the volume of the cone can be calculated using the formula V_cone = (1/3)πr^2h.
Since both the cylinder and the cone share the same radius and height, we can write the volume of the remaining shape as:
V_remaining = V_cylinder - V_cone
= πr^2h - (1/3)πr^2h
= πr^2h - (1/3)πr^2h
Simplifying further, we get:
V_remaining = (2/3)πr^2h
Given that the volume of the cylinder is 54 in^3, we can substitute the value of V_remaining into the equation:
54 = (2/3)πr^2h
To find the volume of the remaining shape (V_remaining), we need to know the value of the radius (r) and height (h). Since the question does not provide that information, we cannot determine the exact volume of the remaining portion. Therefore, none of the given options (27 in^3, 27 in^3, 18 in^3, 18 in^3, 36 in^3, or 36 in^3) are correct.