Anonymous

# Find the maximum possible area of a sector with a perimeter of 20?

K=10r-r^2

then what is the measure of the central angle please explain how answer was arrived at tnx

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Hi,

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

-------.X.πr² =

2πr

(20 - 2r)πr²

---------------

2πr

Cancel out the π and an "r".

(20 - 2r)r

------------ =

2

Multiply the r through the numerator and reduce the fraction.

20r - 2r²

----------- =10r - r² <== This is the formula for the sector's area.

2

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!! :-)

• Area Of A Sector