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Area of a triangle

  1. Dec 6, 2013 #1
    1. The problem statement, all variables and given/known data
    Find the maximum area of a triangle with sides a[itex]\in[/itex] (0,1] ,b[itex]\in[/itex] [1,2], c[itex]\in[/itex] [2,3].


    2. Relevant equations



    3. The attempt at a solution
    I tried to make the area as a function of a single variable so that by differentiating I can get the answer. But it was unsuccessful.
    Thanx in advance.
     
  2. jcsd
  3. Dec 6, 2013 #2

    haruspex

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    Can you determine one of the three lengths immediately?
     
  4. Dec 6, 2013 #3
    Sorry can you elaborate?
     
  5. Dec 6, 2013 #4

    HallsofIvy

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    Since the area of a triangle is given by [tex]\sqrt{s(s- a)(s- b)(s- c)}[/tex] where [tex]s= \frac{a+ b+ c}{2}[/tex] (Heron's formula), does not choosing each of a, b, and c as large as possible, here, (a= 1, b= 2, c= 3), maximize the area?
     
    Last edited by a moderator: Dec 6, 2013
  6. Dec 6, 2013 #5
    Doesn't the greatest side have to be strictly larger than the sum of the two smaller sides?

    Also, if you choose c to be a value like 2.99, that would only give you a tiny sliver of an area as compared to choosing a smaller value.

    Edit: I might have misinterpreted your post. I'm not sure.
     
    Last edited by a moderator: Dec 6, 2013
  7. Dec 6, 2013 #6

    haruspex

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  8. Dec 6, 2013 #7

    haruspex

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    Suppose you had a triangle with no two sides equal, and you were allowed to increase the length of any of them. Which side would you lengthen to be sure of increasing the area?
     
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