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Math Help - Writing the area of a rectangle as a function of an x-coordinate

  1. #1
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    Writing the area of a rectangle as a function of an x-coordinate

    This is the question:

    Express the area of the rectangle shown in the accompanying figure as a function of the x-coordinate of the point P.



    My attempt:

    I know the area of a rectangle is A = xy

    Since it needs to be a function of the x-coordinate, I need to find y in terms of x, so I can substitute it and have only 'x' as a variable. But I'm not sure how to calculate y.

    Any assistance would be great.
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    Hint:

    Compute the area of triangles.
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  3. #3
    MHF Contributor red_dog's Avatar
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    Let A(a,0), \ B(0,b).

    The equation of the line AB is:

    \displaystyle\frac{x-a}{-a}=\displaystyle\frac{y}{b}\Rightarrow y=\frac{b}{a}(a-x)

    The point P is on the line AB, so the coordinates of P satisfy the equation of the line AB.

    Now, replace y in the area of the rectangle:

    A=\displaystyle\frac{b}{a}x(a-x)
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  4. #4
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    Quote Originally Posted by Glitch View Post
    This is the question:

    Express the area of the rectangle shown in the accompanying figure as a function of the x-coordinate of the point P.



    My attempt:

    I know the area of a rectangle is A = xy

    Since it needs to be a function of the x-coordinate, I need to find y in terms of x, so I can substitute it and have only 'x' as a variable. But I'm not sure how to calculate y.

    Any assistance would be great.
    1. The point P is placed on a straight line which passes through the points R(0, b) and S(a, 0).

    2. Determine the equation of the line RS which will provide you with the term of the veariable y.

    3. For confirmation only: I've got y = -\frac ba x + b
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  5. #5
    MHF Contributor Also sprach Zarathustra's Avatar
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    Or:

    ab/2 - {x(b-y)}/2 - {y(a-x)}/2=

    = {ab - xb -xy -ya+yx}/2

    = {ab - ay}/2 =AREA OF RECTANGLE = xy

    ==> ab - ay =xy

    ab=ay+xy

    ==>
    ab= y(a+x)

    y=ab/{a+x}

    so xy= xab/{a+x}
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