A cone-shaped cup is made from a circular piece of paper of radius 12 by cutting out a sector and joining the edges AC and BC.
To find the volume of the cone-shaped cup, we need to know the height of the cone. The height can be determined by the slant height and the radius.
In this case, the radius of the circular piece of paper is given as 12 units. However, we need to find the slant height and the height.
First, let's calculate the slant height. The slant height is the distance from the vertex of the cone (point A) to the edge (point C) along the curved surface of the cone.
To find the slant height, we can use the Pythagorean Theorem, which states that in a right-angled triangle, the square of the hypotenuse (in this case, the slant height) is equal to the sum of the squares of the other two sides.
In our case, the slant height is the hypotenuse, the radius is one side (12 units), and the height is the other side. However, we don't know the height yet.
To calculate the height, we need to use the fact that the curved edge AC is formed by joining the edges of a sector. The length of the curved edge (AC) is equal to the circumference of the circular base of the cone.
The circumference of a circle is given by the formula: Circumference = 2 * π * radius.
In our case, the radius is 12 units, so the circumference of the circular base of the cone is 2 * π * 12 = 24π units.
Now we have the length of the curved edge (AC) as 24π units. We also know that the curved edge forms the slant height of the cone. Therefore, the slant height is 24π units.
Now, we can use the Pythagorean Theorem to find the height. We have the slant height (24π units) and the radius (12 units):
24π^2 = 12^2 + height^2
576π = 144 + height^2
height^2 = 576π - 144
height^2 = 432π
Therefore, the height of the cone-shaped cup is √(432π) units.
Now that we have the height, we can use the formula for the volume of a cone to find the volume of the cone-shaped cup:
Volume = (1/3) * π * radius^2 * height
Radius = 12 units
Height = √(432π) units
Volume = (1/3) * π * 12^2 * √(432π)
Simplifying the equation will depend on the desired level of precision required.