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Math Help - Pieces of Wire

  1. #1
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    Pieces of Wire

    A wire 10 meters long is to be cut into two pieces. One piece will be shaped as an equilateral triangle, and the other piece will be shaped as a circle.

    (a) Express the total area A enclosed by the pieces of wire as a function of the length x of a side of the equilateral triangle.

    (b) Find the domain of A.
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  2. #2
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    Quote Originally Posted by blueridge View Post
    A wire 10 meters long is to be cut into two pieces. One piece will be shaped as an equilateral triangle, and the other piece will be shaped as a circle.

    (a) Express the total area A enclosed by the pieces of wire as a function of the length x of a side of the equilateral triangle.

    (b) Find the domain of A.
    Hello,

    let x be the side of the equilateral triangle
    let y be the perimeter of the circle.

    Then you got:

    3x + y = 10 \Longrightarrow y = 10-3x

    The perimeter of the circle is:
    p = 2 \pi r That means: 2 \pi r = 10 - 3x \Longrightarrow r = \frac{10-3x}{2 \pi}

    Therefore the area of the circle is: A_{circle} = \pi r^2 = \pi \left(\frac{10-3x}{2 \pi} \right)^2 = \frac{(10-3x)^2}{4 \pi}

    The area of an equilateral triangle with side x is:

    A_{\Delta} = \frac{1}{4} x^2 \cdot \sqrt{3}

    Therefore the total area becomes:

    A(x) = \frac{1}{4} x^2 \cdot \sqrt{3} + \frac{(10-3x)^2}{4 \pi}

    (b) Considering the values for x you can see that 0 \leq x \leq \frac{10}{3}
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