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Thread: Rectangle

  1. #1
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    Rectangle

    A rectangle is bounded by the x-axis and the semicircle
    y = sqrt[36 - x^2] . Write the area A of the rectangle as a function of x, and determine the domain of the function.

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  2. #2
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    I assume that the circle is centred at the origin.

    Clearly, its domain is $\displaystyle -6 \leq x \leq 6$ and its range is $\displaystyle 0 \leq y \leq 6$.

    The coordinate of any point on the semicircle is $\displaystyle \left(x, \sqrt{36 - x^2}\right)$.

    Therefore, as you move the point along the semicircle, its length is $\displaystyle x$ in each direction of the horizontal, and it gains a length of $\displaystyle \sqrt{36 - x^2}$ on its vertical.

    So the length of the rectangle is $\displaystyle 2x$ and the width is $\displaystyle \sqrt{36 - x^2}$.

    Therefore its area is

    $\displaystyle A = 2x\sqrt{36 - x^2}$ where $\displaystyle 0 < x< 6$.
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  3. #3
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    Beautifully explained.
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