Max Area of Triangle with Sides (0,1], [1,2], [2,3]

[Vineeth T]

Homework Statement

Find the maximum area of a triangle with sides a$\in$ (0,1] ,b$\in$ [1,2], c$\in$ [2,3].

Homework Equations

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.

 

[haruspex]

Can you determine one of the three lengths immediately?

 

[Vineeth T]

Can you determine one of the three lengths immediately?

Sorry can you elaborate?

 

[HallsofIvy]

Since the area of a triangle is given by $$\sqrt{s(s- a)(s- b)(s- c)}$$ where $$s= \frac{a+ b+ c}{2}$$ (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?

 

[scurty]

Since the area of a triangle is given by $$\sqrt{s(s- a)(s- b)(s- c)}$$ where $$s= \frac{a+ b+ c}{2}$$ (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?

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.

 

[haruspex]

Since the area of a triangle is given by $$\sqrt{s(s- a)(s- b)(s- c)}$$ where $$s= \frac{a+ b+ c}{2}$$ (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?[/QUOTE
No, that would give 0

 

[haruspex]

Sorry can you elaborate?

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?