### Instruction: A circular sheet of paper of radius 6 inches is cut into 3 equal sectors, and each sector is formed into a cone with no overlap. What is the height in inches of the cone?### Response: The first thing I notice is that the problem involves circles and cones, so I expect to use some formulas related to them, such as the area of a circle, the circumference of a circle, and the volume and surface area of a cone.
I also notice that the sheet of paper is cut into 3 equal sectors, which means that each sector has a central angle of 120 degrees and an arc length of one-third of the circumference of the circle.
Now, I imagine forming one of the sectors into a cone.
The arc length of the sector becomes the circumference of the base of the cone, and the radius of the sector becomes the slant height of the cone.
I want to find the height of the cone, which is the perpendicular distance from the vertex to the center of the base.
To find the height of the cone, I need to know the radius of the base of the cone.
I can use the formula for the circumference of a circle to find this: C = 2πr, where C is the circumference and r is the radius.
Since the arc length of the sector is equal to the circumference of the base of the cone, I can set them equal and solve for r:
Arc length of sector = C = 2πr
(1/3) * 2π * 6 = 2πr
4π = 2πr
2 = r
So, the radius of the base of the cone is 2 inches.
Now, I can use the Pythagorean theorem to find the height of the cone, since the slant height, the radius, and the height form a right triangle.
I have:
Slant height = 6
Radius = 2
Height = h
6^2 = 2^2 + h^2
36 = 4 + h^2
32 = h^2
h = √32
h = 4√2
So, the height of the cone is 4√2 inches.