The sector of a with radius 21cm has an area of 280cm^2 if the perimeter of the sector is 68.7 and the sector is bent such that it's straight edges consider to form a cone calculate the vertical angle of the cone

Answer 1

To find the vertical angle of the cone formed by bending the sector, we first need to find the radius and arc length of the sector.

Given:
Radius (r) = 21 cm
Area of sector = 280 cm^2
Perimeter (arc length) of sector = 68.7 cm

To find the radius and arc length of the sector, we can use the following formulas:
Area of sector = (θ/360) * π * r^2
Perimeter of sector = 2πr * (θ/360)

Using the area formula and given values:
280 = (θ/360) * π * 21^2
280 = (θ/360) * π * 441
280 = (θ/360) * 1386.13
θ = (280 * 360) / 1386.13
θ ≈ 72.63

Now, using the arc length formula and given values:
68.7 = 2π * 21 * (72.63/360)
68.7 = 2π * 21 * 0.20175
68.7 = 42.18

Now that we have the radius (21 cm) and arc length (42.18 cm) of the sector, we can calculate the slant height (l) of the cone using the Pythagorean theorem:
l = √(r^2 + (arc length/2)^2)
l = √(21^2 + (42.18/2)^2)
l = √(441 + 892.7082)
l = √1333.7082
l ≈ 36.52 cm

Finally, to find the vertical angle (θ) of the cone, we can use the formula for calculating the cone angle in radians:
θ = 2 * atan(r/l)
θ = 2 * atan(21/36.52)
θ = 2 * atan(0.574)
θ = 76.57 degrees

Therefore, the vertical angle of the cone formed by bending the sector is approximately 76.57 degrees.