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?
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:
Hello, millerst!
This problem requires familiarity with partial derivatives.
Let the three sides be: .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 the square of the area.
That is, use the square of Heron's Formula: .
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.
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
**
We can cancel .
If , then: .
If side is half the perimeter, it violates the Triangle Inequality.
Edit: Nicely done, NonCommAlg!
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