A piece of paper in the form of a sector of a circle of radius 15 cm is rolled into a cone. Calculate the volume of the cone if the angle of the sector is 240 degrees.
To calculate the volume of the cone, we need to know the radius of the cone's base and the height of the cone.
Let's start by finding the radius of the cone's base. The sector of the circle is rolled into a cone, which means the arc of the sector forms the circumference of the cone's base.
The formula to find the circumference of a circle is C = 2πr, where C is the circumference and r is the radius.
Since the arc of the sector is 240 degrees (2/3 of a full circle), we can find the circumference of the base using the formula:
240/360 = C/2πr
Simplifying, we find:
2/3 = C/2πr
Cross-multiplying:
C = (2πr * 2) / 3
C = (4πr) / 3
We know that the circumference of the base is equal to the length of the arc of the sector, which is also equal to the length of the rolled paper. The length of the rolled paper is given by the formula:
L = 2πr * (240/360)
Simplifying the expression:
L = (2πr * 2/3)
Now, we can equate this to the circumference of the base:
(2πr * 2/3) = (4πr) / 3
Simplifying further:
4r/3 = 4r/3
This equation shows that the radius of the base of the cone is not affected by the angle of the sector.
Now that we know the radius of the cone's base is 15 cm, we need to find the height of the cone.
The height of the cone can be found by considering the sector which is rolled to form the cone. We can form a right triangle where the radius of the sector is the hypotenuse of the triangle, and the height of the cone is one of the legs.
The angle of the sector is given as 240 degrees, which means the angle formed in the triangle at the center of the circle is 240/2 = 120 degrees.
Using trigonometry, we can find the height of the cone:
sin(120) = height/radius
sin(120) = height/15
Height = 15 * sin(120)
Height = 15 * (√3/2)
Height = 15√3/2
Now that we know both the radius and the height of the cone, we can calculate the volume using the formula for the volume of a cone:
Volume = (1/3) * π * r^2 * h
Volume = (1/3) * π * 15^2 * (15√3/2)
Volume = (1/3) * π * 225 * (15√3/2)
Volume = (1/3) * π * 225 * (15√3/2)
Volume ≈ 1767.15 cubic centimeters
Therefore, the volume of the cone is approximately 1767.15 cubic centimeters.
Since s=rθ, the circumference of the base of the cone is
15 * 4π/3 = 20π
So, the radius of the base of the cone is 10.
Now you can find the height of the cone, and thus the volume.