Find the maximum possible area of a sector with a perimeter of 20?
then what is the measure of the central angle please explain how answer was arrived at tnx
- Pi R SquaredLv 71 decade agoFavorite Answer
To have a sector with a perimeter of 20, then 2 radii and the arc which is the curved part of the sector must add up to 20. If the radius of the circle is "r". then 2 radii use up "2r" in length, leaving 20 - 2r as the length of the arc along the sector.
Since the circumference of the entire circle is C = 2πr and the sector's arc is 20 - 2r, then the fraction (20 - 2r)/2πr would represent what fractional part of the entire circle would be enclosed in the sector.
Since the area of the entire circle is A = πr², then the area inside the sector would be the fraction for the part of the circle in the sector times the area of the entire circle. This gives a formula for the area of the sector, which is:
(20 - 2r)πr²
Cancel out the π and an "r".
(20 - 2r)r
Multiply the r through the numerator and reduce the fraction.
20r - 2r²
----------- =10r - r² <== This is the formula for the sector's area.
To find the maximum area of the sector, solve for the vertex located on the axis of symmetry. The axis of symmetry is found by the radius "r" = -b/(2a) = -10/(2*-1) = 5. This means the radius is 5 and the arc length is 10.
If the radius is 5, then the circumference is 2π(5) or 10π. The arc on the segment has a length of 10, so the arc is 10/(10π) or .3183, which is 31.83% of the circle. That means the central angle is also 31.83% of 360°, which is (.3183)(360) = 114.588°. This is your central angle.
I hope this helps!! :-)