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Math Help - two problems

  1. #1
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    two problems

    hi there! finally i found a very helpful forum!

    I need two "derivative" solutions of these problems:

    1. from all right-angled triangles whose cathetes sum is 12, find the triangle which has the shortest hypotenuse.

    2. From a rope with length "d" create a rectangle whose surface is maximal.

    thank you with hope that you "can" uderstand the problems, because my english is weak.
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  2. #2
    Senior Member vincisonfire's Avatar
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    (i) Let x and y be the cathete lengths.
    Condition :  x+y=12 \implies y = 12 -x
    H^2 =x^2+y^2 = x^2 + (12-x)^2
     2H\frac{dH}{dx} =2x-2(12-x) = 4x-24 =0 \implies x =6
    If you maximise H^2  you also maximise H
    (ii) Same principle
    Let x be the little side and y be the long side
    Condition :  2x+2y=l \implies y = \frac{l}{2} -x
    S =xy = x\frac{l-2x}{2}
    You differentiate and set equal to 0.
    Use the second derivative test to see the nature of the extrema .
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