ƒ¡ is a circle with center O . A and B are points on ƒ¡ such that the sector AOB has perimeter 40 . What is the measure of ÚAOB (in radians) when the area of the sector AOB is maximized

Question

ƒ¡  is a circle with center O . A  and B  are points on ƒ¡  such that the sector AOB  has perimeter 40 . What is the measure of ÚAOB  (in radians) when the area of the sector AOB  is maximized

Steve · Answer

perimeter = 2r+rθ = 40
r = 40/(θ+2)

a = 1/2 r^2 θ = 800θ/(θ+2)^2
da/dθ = 800(θ-2)/(θ+2)^3

da/dθ = 0 when θ=2 for maximum area
Note that it doesn't really matter what p is; max area for a constant perimeter is when θ=2

ƒ¡ is a circle with center O . A and B are points on ƒ¡ such that the sector AOB has perimeter 40 . What is the measure of �ÚAOB (in radians) when the area of the sector AOB is maximized