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Math Help - Simultaneous Equations: Area of a Rectangle

  1. #1
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    Simultaneous Equations: Area of a Rectangle

    Hello everyone!

    The textbook out of which I am studying provides examples of simultaneous equation problems, but nothing similar to what is being required here. Does anyone know what to do in parts c and d? I have drawn a diagram and done a and b, but I cannot solve the latter two. Much help is appreciated! Thank you in advance to any mathematicians out there.




    A rectangular enclosure is to be fenced so that its perimeter is 60 metres.

    a. If w is the length of the rectangle, what is its length in terms of w?
    l = 30 - w

    b. Write an expression for A, the area of the rectangular enclosure.
    A = w(30 - w)

    c. What value of w will give the greatest area of the rectangular enclosure?

    d. What is the greatest area the rectangular enclosure can have?

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  2. #2
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    Do you know how to find the derivative of A with respect to w?
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  3. #3
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    No, I'm afraid not. I am only in Year 9. Is there any other possible way?
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  4. #4
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    I agree with your a) and b).

    You have a Quadratic equation for the area. So you should be able to put it in turning point form, so you can read off the maximum.

    A = w(30 - w)

    A = 30w - w^2

    A = -(w^2 - 30w)

    A = -\left[w^2 - 30w + (-15)^2 - (-15)^2\right]

    A = -\left[(w - 15)^2 - 225\right]

    A = -(w - 15)^2 + 225.


    Therefore the turning point is (15, 225). Since this is a "negative" quadratic, the turning point is a maximum. Therefore, the maximum area is 225\,\textrm{m}^2 and occurs when the width is 15\,\textrm{m}.
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  5. #5
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    Note: Prove It used "completing the square".
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