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Math Help - Cutting Wire into Two Pieces

  1. #1
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    Cutting Wire into Two Pieces

    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) What is the domain of the area A?

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  2. #2
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    Hello, magentarita!

    A wire 10 meters long is to be cut into two pieces. One piece will be bent
    into an equilateral triangle and the other piece will be bent into 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.
    Let x = side of the triangle.
    The area of an equilateral triangle of side x is: . A_t \:=\:\frac{\sqrt{3}}{4}x^2


    The triangle uses 3x of the wire.
    This leaves 10-3x for the circumference of the circle.

    Circumference formula: . C \:=\:2\pi r

    So we have: . 10-3x \:=\:2\pi r \quad\Rightarrow\quad r \:=\:\frac{10-3x}{2\pi}

    Area formula: . A \:=\:\pi r^2

    So we have: . A_c \;=\;\pi\left(\frac{10-3x}{2\pi}\right)^2 \quad\Rightarrow\quad A_c\;=\;\frac{(10-3x)^2}{4\pi}


    Therefore: . \boxed{A \;=\;\frac{\sqrt{3}}{4}x^2 + \frac{(10-3x)^2}{4\pi}}



    (b) What is the domain of A ?

    Domain: . 0 \,\leq\,x\,\leq\,3\tfrac{1}{3}


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  3. #3
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    Soroban....

    Quote Originally Posted by Soroban View Post
    Hello, magentarita!

    Let x = side of the triangle.
    The area of an equilateral triangle of side x is: . A_t \:=\:\frac{\sqrt{3}}{4}x^2


    The triangle uses 3x of the wire.
    This leaves 10-3x for the circumference of the circle.

    Circumference formula: . C \:=\:2\pi r

    So we have: . 10-3x \:=\:2\pi r \quad\Rightarrow\quad r \:=\:\frac{10-3x}{2\pi}

    Area formula: . A \:=\:\pi r^2

    So we have: . A_c \;=\;\pi\left(\frac{10-3x}{2\pi}\right)^2 \quad\Rightarrow\quad A_c\;=\;\frac{(10-3x)^2}{4\pi}


    Therefore: . \boxed{A \;=\;\frac{\sqrt{3}}{4}x^2 + \frac{(10-3x)^2}{4\pi}}



    Domain: . 0 \,\leq\,x\,\leq\,3\tfrac{1}{3}

    You are fantastic. I love your replies. I am going to take calculus 1 soon. Are you interested in helping me as I go through calculus 1?
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