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)
Responses
40 in.3
40 in. cubed
27 in.3
27 in. cubed
36 in.3
36 in. cubed
18 in.3
The volume of the cylinder before removing the cone is 54 in^3. Since the cone and cylinder have the same radius and height, the volume of the cone is equal to 1/3 of the volume of the cylinder.
To find the volume of the cone, we use the formula for the volume of a cone: V = (1/3)πr^2h, where r is the radius of the cone and h is the height of the cone.
Since the cone and cylinder have the same radius and height, we can call both r and h as 'x'.
So, the volume of the cone is V = (1/3)πx^2x = (1/3)πx^3.
The volume of the remaining amount after removing the cone from the cylinder is the difference between the volume of the cylinder and the volume of the cone.
Volume remaining = Volume of cylinder - Volume of cone
Volume remaining = 54 in^3 - (1/3)πx^3
We don't have a specific value for 'x', but we do have the expression (1/3)πx^3. We can't simplify this further, so we leave our answer in terms of this expression.
Therefore, the correct response is: 54 in^3 - (1/3)πx^3
wrong
Apologies for the incorrect response.
Given that the cone and cylinder share the same radius and height, the volume of the remaining amount after removing the cone can be calculated by subtracting the volume of the cone from the volume of the cylinder.
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 and cylinder have the same radius and height, let's call them both 'r' and 'h'.
The volume of the cylinder is V_cylinder = πr^2h.
The volume of the cone is V_cone = (1/3)πr^2h (since it is one-third of the volume of the cylinder).
Therefore, the volume of the remaining amount is:
Volume remaining = V_cylinder - V_cone
= πr^2h - (1/3)πr^2h
= (2/3)πr^2h
We are given that the volume of the cylinder before removing the cone is 54 in^3. So we can set up the equation:
(2/3)πr^2h = 54
We are not given a specific value for the radius or height, so we cannot determine the exact volume of the remaining amount. Therefore, we cannot choose a specific response option.
The correct answer is that the volume of the remaining amount is (2/3)πr^2h.