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Math Help - Optimization problem

  1. #1
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    Optimization problem

    i have a question about optimization so this is the question i was given "the figure below consists of a rectangle ABCD and two semicircles on either end. the rectange as an area of 100 cm^2. If x represents the length of the rectangle AB, find the value of x that makes the perimeter of the entire figure a minimum."

    so i did all the steps but i just don't get what im doing wrong... here is what i did

    1) Area = x times w,
    expressing w in terms of x i get w=100/x

    2) the total perimeter will be the sum of the rectangle perimeter and the circle perimeter... the diameter of the circle is the same as the width...

    3) plugging all known information i get,
    P(total) = 200/x + 2x + 100pi/x

    4) differentiating the function gives,
    P'(total)= -200/x^2 + 2 - 100pi/x^2

    5) solving for x i get x equal to 16.034 cm... i know that using the second derivative it gives that its a minimum point but for some reason the answer in the book is 12.5 cm... what did i do wrong here?
    Optimization problem-untitled.png
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Optimization problem

    For your perimeter, you want:

    P(x)=2x+\frac{100\pi}{x}

    You added the sides of the rectangle that don't contribute to the perimeter of the figure.
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  3. #3
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    Re: Optimization problem

    Hey bakerkhojah.

    For the perimeter of the two circles, the perimeter of one semi-circle will be pi*radius and if you choose x as the diameter then the semi-circle will have a radius of pi*x/2 which means two semi-circles will give a perimeter of a full circle which is pi*x.

    If you let the other side have the semi-circles, the perimeter will be 100*pi/x.

    But no matter what side you pick, you will always leave out the term for the perimeter that you use for the semi-circles.

    So if you use pi*x for the circle perimeter you leave out 2*x in the perimeter calculation and if you use 100*pi/x for the circle perimeter you leave out 200/x for the perimeter calculation because the perimeter is based on the semi-circles only and not on the rectangle (since this is always inside the shape and not part of the perimeter).
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