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.
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
3 answers
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.
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.
1 answer