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Math Help - optimization question

  1. #1
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    optimization question

    please help me with thus questions.

    1) A rectangle study area is to be enclosed by a fence and divided into two equal parts, with a fence running along the division parallel to one of the sides. If the total area is 384ft^2, find the dimenstions of the study area that will minimize the total length of the fence. How much fence will be required?

    2) Show that among all rectangles with a given perimeter, the square has the largest area.

    3) How close does the curve y=1/x come to the origin?

    thank you. Any help would be appreciated.
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  2. #2
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    Quote Originally Posted by inhae772 View Post
    please help me with thus questions.

    1) A rectangle study area is to be enclosed by a fence and divided into two equal parts, with a fence running along the division parallel to one of the sides. If the total area is 384ft^2, find the dimenstions of the study area that will minimize the total length of the fence. How much fence will be required?

    2) Show that among all rectangles with a given perimeter, the square has the largest area.

    3) How close does the curve y=1/x come to the origin?

    thank you. Any help would be appreciated.
    1. let F = total length of fence

    F = 2L + 3W

    LW = 384 ... L = \frac{384}{W}

    F = 2\left(\frac{384}{W}\right) + 3W

    find \frac{dF}{dW} and minimize


    2. P = 2L + 2W , P is a fixed constant

    L = \frac{P-2W}{2}

    A = LW

    A = \frac{P-2W}{2} \cdot W

    find \frac{dA}{dW} and maximize


    3. point on the curve ... \left(x , \frac{1}{x}\right)

    origin (0,0)

    distance formula ...

    D = \sqrt{(x-0)^2 + \left(\frac{1}{x} - 0\right)^2}

    D = \sqrt{x^2 + \frac{1}{x^2}}

    shortcut ...
    if you minimize x^2 + \frac{1}{x^2} , then you also minimize the distance, D
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