For a given perimeter, what type of triangle encloses the most area.

How do I do this since triangles have three sides, a b and c for perimeter?

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- Apr 6th 2009, 01:57 PMmillerstOptimization
For a given perimeter, what type of triangle encloses the most area.

How do I do this since triangles have three sides, a b and c for perimeter? - Apr 6th 2009, 02:45 PMNonCommAlg
there are so many ways to solve this problem. i'll give you two of them. i'll assume that are the length of the sides and where is fixed.

__Solution 1__: by Heron's formula: so we only need to maximize the function

so we must solve theis system: which will easy give you and thus so the equilateral

triangle with the side of length is the answer.

__Solution 2__: following our notation in solution 1, by AM-GM inequality we have: which gives us:

- Apr 6th 2009, 04:57 PMSoroban
Hello, millerst!

This problem requires familiarity with partial derivatives.

Quote:

For a given perimeter, what type of triangle encloses the most area?

The perimeter is a constant,

We can use Heron's Formula: .

. . where is the semiperimeter: .

To maximize the area, we can maximize theof the area.*square*

That is, use theof Heron's Formula: .*square*

We find that: .

Then [1] becomes: .

which simplifies to: .

Equate the partial derivatives to zero:

. .

We have: .

Solve [5] for

. . .**

Substitute into [4]: .

. . which simplifies to: .

Back-substitute and we get: .

As you probably suspected, the triangle of maximum area is.*equilateral*

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

**

Wecancel .__can__

If , then: .

If side is*half the perimeter*, it violates the Triangle Inequality.

Edit: Nicely done, NonCommAlg!

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