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Math Help - calc 3, maximize the area

  1. #1
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    calc 3, maximize the area

    a triangular field is to be enclosed by p feet of fencing so as to maximize the area of the field. find the lengths of sides of this triangle. [Hint: Heron's formula for the area of a triangle with side lengths x, y, and z is A= square root of (s(s-x)(s-y)(s-z)), where s= 1/2 (x+y+z) is the semiperimeter.]

    so, there's a function with two constraints.
    i took the gradient of each.
    i set gradient p = lambda gradientA + mu gradientS
    this is extremely complicated though and i don't know how to solve it.

    the back of the book says x=y=z=p/3

    any suggestions? thank you so much
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by holly123 View Post
    a triangular field is to be enclosed by p feet of fencing so as to maximize the area of the field. find the lengths of sides of this triangle. [Hint: Heron's formula for the area of a triangle with side lengths x, y, and z is A= square root of (s(s-x)(s-y)(s-z)), where s= 1/2 (x+y+z) is the semiperimeter.]

    so, there's a function with two constraints.
    i took the gradient of each.
    i set gradient p = lambda gradientA + mu gradientS
    this is extremely complicated though and i don't know how to solve it.

    the back of the book says x=y=z=p/3

    any suggestions? thank you so much
    Here's a hint: maximizing \sqrt{s(s-x)(s-y)(s-z)} is the same as maximizing s(s-x)(s-y)(s-z). That should make the calculations simpler.
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