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Math Help - Trig tent question

  1. #1
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    achille
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    Trig tent question

    Okay, so this problem goes like this: A two-person tent is to be made so that the height at the center is a = 4 feet (see the figure below). If the sides of the tent are to meet the ground at an angle 60, and the tent is to be b = 5 feet in length, how many square feet of material will be needed to make the tent? (Assume that the tent has a floor and is closed at both ends, and give your answer in exact form.) I am not sure how to set this one up. It looks like an equilateral triangle.


    Trig tent question-1-1-057-alt.gif
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  2. #2
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    Re: Trig tent question

    Hello, goldbug78!

    A two-person tent is to be made so that the height at the center is a = 4 feet.
    If the sides of the tent are to meet the ground at an angle 60, and the tent is to be b = 5 feet in length,
    how many square feet of material will be needed to make the tent?
    (Assume that the tent has a floor and is closed at both ends, and give your answer in exact form.)

    Code:
                            *
                        *    \
                    *         \
                *              \
               /|\              \
              / | \              \
           x /  |4 \ x            *
            /   |   \         *
           /    |    \    * 5
          *-----*-----*
          : x/2 : x/2 :
    The front is an equilateral triangle with altitude 4 feet.

    Pythagorus: . \left(\tfrac{x}{2}\right)^2 + 4^2 \:=\:x^2 \quad\Rightarrow\quad \tfrac{x^2}{4} + 16 \:=\:x^2

    . . . . . . . . . . \tfrac{3}{4}x^2 \:=\:16 \quad\Rightarrow\quad x^2 \:=\:\tfrac{64}{3} \quad\Rightarrow\quad x \:=\:\tfrac{8\sqrt{3}}{3}


    The front triangle has area: . \tfrac{1}{2}\left(\tfrac{8\sqrt{3}}{3}\right)(4) \:=\:\tfrac{16\sqrt{3}}{3}
    . . The two triangular panels have area: . 2 \times \tfrac{16\sqrt{3}}{3} \:=\:\tfrac{32\sqrt{3}}{3}

    A rectangular panel has area: . \left(\tfrac{8\sqrt{3}}{3}\right)(5) \:=\:\tfrac{40\sqrt{3}}{3}
    . . The three rectangular panels have area: . 3 \times\tfrac{40\sqrt{3}}{3} \:=\: 40\sqrt{3}

    The total area is: . \frac{32\sqrt{3}}{3} + 40\sqrt{3} \;=\;\frac{152\sqrt{3}}{3}\text{ ft}^2
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