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Thread: Trig Optimization Problem

  1. #1
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    Trig Optimization Problem

    I am having a bit of trouble getting started with this problem:
    The hypotenuse of a right-angled triangle is 12 cm in length. Find the measures of the angles in the triangle that maximize its perimeter.
    Any help would be appreciated.
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  2. #2
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    calculus

    It's a right triangle. So x^2 + y^2 = 12^2. You want to maximize the perimenter = x+y+12. After you have x and y, you can use trig to find the angles
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  3. #3
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    Re: Trig Optimization Problem

    Right Angle Triangle

    Perimeter = Side1 + S2 + S3

    Side 1 = 12cm (hypotenuse)

    S2 = 12cosX (if you draw a right angled triangle you can figure this out through basic trig)
    S3 = 12sinX

    Therefore;

    P(x) = Side1 + S2 + S3
    P(x) = 12 + 12CosX + 12SinX

    To find max; use calculus!

    P'(x) = 12(-sinX) + 12(cosX)
    P'(x) = 12(cosX - sinX)

    Max when this = 0

    P'(x) = 0
    12(cosX - sinX) = 0
    therefore; cosX - sinX = 0
    therefore; cosX = sinX

    Only place these are = to eachother is at 45 degrees, or Pi/4 radians.

    To check this is a max. Upon graphing 0 < x < 90 (angle cannot be less than or zero, or cannot be more than or equal to 90 degrees)

    Graph

    0_______ (positive)_________45 (0)_______________(negative)_____________90

    transitions from + to - therefore Perimeter is maximised at 45 degrees.

    Hope this helped
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