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Math Help - Trig problem I'm stuck on

  1. #1
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    Trig problem I'm stuck on

    I have a triangle that has internal angles of 35, 80 and 65 degrees. The area of the triangle needs to be 4000sq metres. Say I need to fence this triangle. Will 285 metres of fencing material be enough? If this is not enough fencing material can I change the side lengths of the triangle such that the area is still 4000sq m and no more than 285m of material is used? I'm not able to change the angles in this triangle by more than +or- 25 degrees.
    Even just a suggestion as to where to start tackling this problem would be greatly appreciated.
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  2. #2
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    Hello Eleven Eleven
    Quote Originally Posted by Eleven Eleven View Post
    I have a triangle that has internal angles of 35, 80 and 65 degrees. The area of the triangle needs to be 4000sq metres. Say I need to fence this triangle. Will 285 metres of fencing material be enough? If this is not enough fencing material can I change the side lengths of the triangle such that the area is still 4000sq m and no more than 285m of material is used? I'm not able to change the angles in this triangle by more than +or- 25 degrees.
    Even just a suggestion as to where to start tackling this problem would be greatly appreciated.
    I don't think you can quite do this with 285 m - the mimimum length of fencing you need will be 288 m.

    Here's the calculation.

    Suppose the triangle is ABC with A = 35, B = 80, C = 65.

    The area of the triangle is \Delta= \tfrac12ab\sin C, and using the Sine Rule b = \frac{a\sin B}{\sin A}. So
    \Delta = \frac12\frac{a^2\sin B \sin C}{\sin A}

    \Rightarrow a^2 = \frac{2\Delta\sin A}{\sin B \sin C}

    =\frac{8000 \sin 35}{\sin 80\sin65}

    \Rightarrow a = 71.70
    This gives b = \frac{a\sin B}{\sin A}=123.11 and c=\frac{a\sin C}{\sin A}=113.29. The perimeter is then 308.1 m.

    The minimum perimeter for a given area will be when the triangle is equilateral, all angles then being 60^o. But this gives each side as 96.11 m, with a perimeter of just over 288 m.

    Grandad
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  3. #3
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    Thanks

    That's great. It's got me started so thank you!
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