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)
The volume of a cylinder is given by V = πr^2h, where r is the radius and h is the height.
The volume of the cone removed from the cylinder is given by V_cone = (1/3)πr^2h.
Since the cone and cylinder have the same radius and height, the volume of the cone equals (1/3) times the volume of the cylinder.
Therefore, the volume of the cone is V_cone = (1/3)(54) = 18 in^3.
So, the volume of the remaining amount is V_remaining = V_cylinder - V_cone = 54 - 18 = <<54-18=36>>36 in^3. Answer: \boxed{36}.
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 18 in. 3 18 in. cubed 36 in. 3 36 in. cubed 40 in. 3 40 in. cubed 27 in. 3 chose one of these
The volume of the remaining amount is 36 in.³
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 18 in. 3 18 in. cubed 36 in. 3 36 in. cubed 40 in. 3 40 in. cubed 27 in. 3
The volume of the remaining amount is 36 in.³
To find the volume of the amount remaining after removing the cone from the cylinder, we first need to find the volume of the cylinder and the volume of the cone.
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius of the cylinder and h is the height of the cylinder.
Since the cone and the cylinder have the same radius and height, let's call the radius and height of both the cylinder and the cone as r and h, respectively.
Given that the volume of the cylinder before removing the cone is 54 in^3, we can now set up the equation:
54 = πr^2h
Next, let's find the volume of the cone. The volume of a cone is given by the formula V = (1/3)πr^2h.
Since the cone and the cylinder have the same radius and height, the volume of the cone is:
V_cone = (1/3)πr^2h
To find the volume of the remaining amount, we subtract the volume of the cone from the volume of the cylinder:
V_remaining = V_cylinder - V_cone
Substituting the respective formulas for the cylinder and the cone:
V_remaining = πr^2h - (1/3)πr^2h
Factoring out πr^2h, we have:
V_remaining = πr^2h(1 - 1/3)
Simplifying the expression inside the parentheses:
V_remaining = πr^2h(2/3)
Now, substituting the given volume of the cylinder (54 in^3) into the equation:
54 = πr^2h
We can solve for πr^2h:
πr^2h = 54
Rearranging the equation:
r^2h = 54/π
Substituting this value into the volume equation for the remaining amount:
V_remaining = πr^2h(2/3)
V_remaining = π(54/π)(2/3)
Canceling out the π terms:
V_remaining = (54/π)(2/3)
Finally, simplifying the expression:
V_remaining = 36 in^3
Therefore, the volume of the remaining amount after removing the cone from the cylinder is 36 in^3.