A sector of a circle of radius 8cm is bent to form a cone. Find the radius of the cone and it vertical angle if the angle sub tended at the center by the sector is 280°
radius of cone (r) ... r = (280º / 360º) * 8 cm
vertical angle (Θ) ... sin(Θ / 2) = r / 8 cm
Help me to solve it
2.17
To find the radius of the cone formed from a sector of a circle, we need to use the formula for the circumference of the circle that the sector comes from.
The formula for the circumference of a circle is C = 2πr, where C is the circumference and r is the radius. Since we know the angle subtended at the center by the sector is 280°, we can find the fraction of the entire circumference that this angle represents.
To find this fraction, we use the following formula:
Fraction of circumference = (Angle subtended / 360°)
In this case, the fraction of the circumference represented by the sector is (280° / 360°) = 7/9.
Since the sector is bent to form a cone, this fraction of the circumference will also represent the fraction of the base circumference of the cone. Therefore, we can set up the following equation:
(7/9) * (2πr) = 2πR
In this equation, R represents the radius of the cone. Simplifying the equation gives us:
(7/9) * r = R
Now, we know the radius of the sector is 8 cm, so we can substitute this value into the equation:
(7/9) * 8 = R
Solving for R, we get:
R = (7/9) * 8
= 56/9
≈ 6.22 cm
The radius of the cone is approximately 6.22 cm.
To find the vertical angle of the cone, we can use the following formula:
Vertical angle = 360° * (Fraction of circumference)
In this case, the fraction of the circumference represented by the sector is 7/9. So, the vertical angle of the cone is:
Vertical angle = 360° * (7/9)
= 252°
Therefore, the vertical angle of the cone is 252°.