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
To find the volume of the remaining amount after carving out the cone, we can subtract the volume of the cone from the volume of the cylinder.
The volume of a cylinder is given by the formula V = πr²h, where r is the radius and h is the height.
The volume of a cone is given by the formula V = (1/3)πr²h, where r is the radius and h is the height.
Since the cone and cylinder have the same radius and height, we can let r = h = x.
The volume of the cylinder is then Vcylinder = πx²x = πx³.
Given that the volume of the cylinder is 54 in^3, we can set up the equation πx³ = 54.
To find the value of x, we can solve for it:
x³ = 54/π
x = (54/π)^(1/3)
Now, we can find the volume of the cone by plugging in x:
Vcone = (1/3)πx²x = (1/3)π[(54/π)^(1/3)]²[(54/π)^(1/3)].
Finally, we subtract the volume of the cone from the volume of the cylinder:
Vremaining = Vcylinder - Vcone = πx³ - (1/3)πx²x.
Simplifying the expression, we have:
Vremaining = (2/3)πx³.
Plugging in the value of x, we have:
Vremaining = (2/3)π[(54/π)^(1/3)]³.
Calculating this expression will give us the volume of the remaining amount.
Okay so
we can simplify the volume of the remaining amount calculation:
The volume of the remaining amount (Vremaining) is equal to the volume of the cylinder (Vcylinder) minus the volume of the cone (Vcone):
Vremaining = Vcylinder - Vcone.
Since the cylinder and cone have the same radius and height, we can simplify this as:
Vremaining = πr²h - (1/3)πr²h,
where r is the radius and h is the height.
The volume of the cylinder before removing the cone is given as 54 in^3, so we can substitute this in:
Vremaining = 54 - (1/3)πr²h.
Since we don't know the exact values of r and h, we can't calculate the precise volume of the remaining amount. However, we can simplify the expression by noting that (1/3)πr²h represents the volume of the cone.
Therefore, the volume of the remaining amount is 54 minus the volume of the cone, which is 54 - Vcone.
In conclusion, without knowing the specific values of the radius and height, we cannot determine the exact volume of the remaining amount, but it can be expressed as 54 - Vcone.
To find the volume of the remaining material after removing the cone, we need to subtract the volume of the cone from the volume of the cylinder.
1. First, let's find the volume of the cylinder using the formula V = πr^2h, where V is the volume, r is the radius, and h is the height. We are given that the volume of the cylinder is 54 in.3, so we can set up the equation as 54 = πr^2h.
2. Since the cone and the cylinder have the same radius and height, we can assume that the radius and height of the cone are also r and h, respectively.
3. The formula for the volume of a cone is V = (1/3)πr^2h. Since the cone is carved out of the cylinder, the volume of the cone will be subtracted from the volume of the cylinder. So we need to subtract (1/3)πr^2h from 54 to find the volume of the remaining material.
4. Simplifying the equation, we have 54 - (1/3)πr^2h = remaining volume.
To determine the specific volume of the remaining material, we need to know the value of π (pi) and the numerical values of the radius and height. The possible options mentioned in the question only provide the answer choices and not the specific values of π, r, and h. Therefore, to find the correct answer, we would need more information from the question or solve for the remaining volume using known values of π, r, and h.