precalc question

Question

Determine the angle needed for a sector whose perimeter is four times as big as the area.The radius of the sector is 3 feet.

Answer 1

Area of a circle = pi(r)^2 = 9pi if r=3

Area of a sector is a percentage of the circle

For a circle the angle is 2pi, and area = 2pi/2(r)^2

For a sector the area = angle/2 (r)^2

Perimeter or arc length = angle times radius with angle measured in radians = 3 times the angle when radius =3

Perimeter or arc length or circumference = 2rpi = radius times 2pi if the angle =2pi

arc length of a sector is a percentage of the circumference of a circle = angle/2pi times radius = 3angle/2pi

If perimeter or arc length = 4 times area, then

3(angle)/2pi = 4(angle/2)(r)^2

(3/2)angle/pi = 2(angle)(9)

(1/2)angle/pi = 6(angle)

angle cancels leaving

pi/2 = 6 which is not true

Only angles that would satisfy the requirement are

angle of sector = 0 or infinity

with an angle of 0, the arc length and area are 0, and 0 = 4 times 0

same with an angle of infinity. Infinity times 3 = any other multiple of infinity

Answer: the sector angle = 0

as infinity also could be a solution, it is not a finite number, while zero is.

Set the arc length = 4 times the area

0 times radius = 4 times 0 times anything

0 = 0