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
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.