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How do i do this maths question??

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A sector of a circle with radius r cm contains an angle of θ radians between the bounding radii. Given that the perimeter of the sector is 7cm, express θ in terms of r and show that the area is r/2(7-2r)cm square. Hence, find the maximum area of this sector as r varies.

I have already done the first part of the question, i only don't know how to do the "hence...". Please help, Thanks!

  • How do i do this maths question?? -

    arc length = a = rѲ
    Then P = 2r + rѲ
    7 = 2r + rѲ
    Ѳ = (7-2r)/r

    Let the area of the sector be A
    by ratios
    A/(pir^2) = Ѳ/(2pi)
    A = (r^2)Ѳ/2
    or
    Ѳ = 2A/r^2

    then 2A/r^2 = (7-2r)/r
    solving for A after cross-multiplying and simplifying
    A = (7r-2r^2)/2 or r/2(7-r) as required

    second part:
    A = (7/2)r - r^2 from above
    dA/dr = 7/2 - 2r
    = 0 for a max/min of A
    2r - 7/2 = 0
    r = 7/4
    so Ѳ = (7 - 2(7/4))/(7/4)
    Ѳ = 2 (how nice)

    so max area = (r^2)(Ѳ)/2
    = (49/16)(2)/(7/4)
    = 7/2

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