a 120 degree sector is cut out of a circular piece of a Tin with radius of 6 in. and vent to form the lateral surface of a cone. what is the volume of a cone

Question

can someone help me figure this out? any help is appreciated, thanks!

Answer 1

All good with the above answer except

V = 1/3πr²h

V = 1/3π(4²)(4.47)

Answer 2

Volume of a cone is:

 

V = 1/3πr²h

 

We'd like to figure out the cone's radius and height. We know that the cone's slant height is 6 in., since the radius of the circular piece of tin becomes the height of the cone. If we found radius, we would be able to use it and the slant height to find the cone's height with the Pythagorean Theorem. So, let's work on figuring out the radius.

 

The piece of tin had a circumference of:

 

C = 2πr

C = 2π(6)

C = 12π inches

 

We've cut a 120° sector out, which is 120/360 = 1/3 of the circle's circumference. That leaves us with 2/3 of the circumference to make up the cone's base's circumference. 2/3 of 12π inches is (2/3)12π = 8π inches.

 

Since the cone's circumference is 8π inches, we can find its radius:

 

C = 2πr

8π = 2πr

8 = 2r

4 = r

 

The cone's radius is 4 inches.

 

Let's use that and the slant height to find the cone's height with the Pythagorean Theorem:

 

6² = 4² + h²

36 = 16 + h²

20 = h²

4.47 = h

 

 

The height of the cone is 4.47 inches. (Are you sure the slant height is 6 inches and not 5 inches, by the way? That would change things a bit.) That means the volume of the cone is:

 

V = 1/3πr²h

V = 1/3π(4)(4.47)

V = 5.96π

V ≈ 18.72 cubic inches