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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

  • Geometry -

    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

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