a piece of paper in the shape of sector of circle is rolled up to form a right circular cone. what is the measure of the angle theta if its height is twelve centimeter and radius is 5 cms
As always, draw a picture.
The radius of the circle is the slant height of the cone.
The arc length of the paper is the circumference of the cone.
So, the cone's slant height is 13
The circumference of the cone is 10π
So, the angle θ of the sector is found using
s = rθ
10π = 13θ
To find the measure of the angle theta, we need to use the properties of a sector of a circle and relate it to the right circular cone.
Let's begin by understanding the relationship between a sector of a circle and a cone.
When a sector of a circle is rolled up to form a cone, the radius of the circle sector becomes the slant height of the cone, and the angle at the center of the sector becomes the angle at the vertex of the cone.
In this case, we are given that the height of the cone is 12 cm and the radius is 5 cm. We need to find the measure of the angle theta.
Step 1: Find the slant height of the cone.
The slant height of the cone is equal to the radius of the circle sector. Hence, the slant height is 5 cm.
Step 2: Find the circumference of the base of the cone.
The circumference of the base of the cone is equal to the circumference of the circle sector. The circumference of a circle is given by the formula: Circumference = 2πr.
Since the radius of the circle sector is 5 cm, the circumference of the base of the cone is 2π(5) = 10π cm.
Step 3: Use the properties of a right circular cone to find the angle theta.
The slant height of a cone, the radius of the base, and the angle between them form a right triangle. We can use the trigonometric function sine to find the angle theta.
In this case, sine(theta) = Opposite/Hypotenuse.
Using the right triangle formed by the height, the slant height, and the radius (hypotenuse), we have:
sine(theta) = height/slant height
sine(theta) = 12/5
To find the value of theta, we take the inverse sine (or arcsine) of both sides.
theta = arcsine(12/5)
Using a calculator, we can solve for theta.
theta ≈ 66.80 degrees (rounded to two decimal places)
Therefore, the measure of angle theta is approximately 66.80 degrees.