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?

Answer 1

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 carved out has the same radius and height as the cylinder, its volume can be calculated using the same formula.

The volume of the cone is given by V_cone = (1/3)πr^2h.

The volume of the amount remaining is the difference between the volume of the cylinder and the volume of the cone:
V_remaining = V_cylinder - V_cone.

Substituting the formulas for V_cylinder and V_cone, we have:
V_remaining = πr^2h - (1/3)πr^2h.

Factoring out the common factor of πr^2, we get:
V_remaining = πr^2h (1 - 1/3).

Simplifying the expression, we have:
V_remaining = πr^2h (2/3).

Given that the volume of the cylinder before removing the cone is 54 in^3, we can substitute this value in for V_cylinder.

54 = πr^2h.

Solving for h, we have:
h = 54 / (πr^2).

Substituting this value of h into the expression for V_remaining, we get:
V_remaining = πr^2 (54 / (πr^2)) (2/3).

Simplifying further, we have:
V_remaining = (54 / 3) (2/3).

Multiplying the numerators, we have:
V_remaining = 108 / 9.

Dividing 108 by 9, we get:
V_remaining = 12.

Therefore, the volume of the amount remaining is 12 in^3.

Answer 2

36 in.336 in. cubed27 in.327 in. cubed40 in.340 in. cubed18 in.3