A 240° sector of a circle of radius 14cm is bent to form a cone.find the radius of the base of the cone, and it vertical angle.
Answer the question please
To find the radius of the base of the cone, we can start by considering the circumference of the original circle.
The entire circle has a circumference of 2πr, where r is the radius.
Given that the sector is 240°, it represents 240/360 = 2/3 of the entire circle.
Therefore, the circumference of the sector is (2/3) * 2πr.
Since the circumference of the sector is equal to the base of the cone, we can equate these two values:
(2/3) * 2πr = 2πR
Where R is the radius of the base of the cone.
By canceling out 2π on both sides of the equation, we get:
(2/3) * r = R
Now, let's substitute the given values.
r = 14 cm (radius of the original circle)
Substituting this value into the equation, we find:
(2/3) * 14 = R
(28/3) = R
R ≈ 9.33 cm
Therefore, the radius of the base of the cone is approximately 9.33 cm.
Next, let's find the vertical angle of the cone.
The formula to find the vertical angle of the cone, θ, is:
θ = (360° * R) / (π * r)
Substituting the given values, we get:
θ = (360° * 9.33) / (π * 14)
θ = 334.8° / (π * 14)
θ ≈ 7.5°
Therefore, the vertical angle of the cone is approximately 7.5°.
To find the radius of the base of the cone, we need to find the circumference of the circle.
The formula for circumference is:
Circumference = 2πr,
where π (pi) is approximately 3.14159.
We have a sector of the circle that covers an angle of 240°. The total angle of a circle is 360°.
To find the circumference of the sector, we can use the formula:
Circumference of the sector = (angle of the sector / total angle of the circle) * Circumference of the circle.
Given that the radius of the circle is 14cm, the formula becomes:
Circumference of the sector = (240 / 360) * (2π * 14).
Simplifying this expression:
Circumference of the sector = (240 / 360) * (2 * 3.14159 * 14) = 9.4247 * 14 = 131.9468 cm.
The circumference of the sector is equal to the base circumference of the cone. Therefore, the radius of the base of the cone is equal to the circumference of the sector divided by 2π.
Radius of the base of the cone = 131.9468 / (2 * 3.14159) = 20.9925 cm (approx).
To find the vertical angle of the cone, we can use the following formula:
Vertical angle = (angle of the sector / total angle of the circle) * 360°.
Given that the angle of the sector is 240°, we can calculate the vertical angle as:
Vertical angle = (240 / 360) * 360° = 240°.
Therefore, the radius of the base of the cone is approximately 20.9925 cm and the vertical angle is 240°.
Make a sketch to see that the arc length of the sector becomes the circumference of the circular base of the cone, and the radius of the sector
becomes the slant height of the cone.
arc length = (240/36)(2π)(14) = 56/3 π cm
so for radius R of the cone's base:
2πR = 56/3 π
R = 28/3 cm
Height of cone, call it h:
h^2 + (28/3)^2 = 14^2
h = ....
tan(half of vertical angle) = 14/(28/3) = 3/2
half of vertical angle = ....
vertical angle = .....