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### Triangle Area

```
Date: 05/24/99 at 11:22:18
From: Dermot
Subject: Calculus

A right triangle has a perimeter of 12m. Find the maximum area of the
triangle.
```

```
Date: 05/24/99 at 13:43:32
From: Doctor Jaffee
Subject: Re: Calculus

Hi Dermot,

This problem looks a lot easier than it is, but I think I can help you
solve it. Here is the outline of my solution.

Let x and y be the lengths of the two legs of the triangle. Since the
perimeter is 12, the length of the hypotenuse is 12 - (x + y).

Furthermore, according to the Pythagorean theorem,

x^2 + y^2 = [12 - (x + y)]^2

You can simplify this expression and solve for y in terms of x.

Now, since area = (1/2)*base*height, substitute x for the base and
your value of y in terms of x for the height.

You want to maximize the area, so find the derivative dA/dx, set it
equal to zero, and solve for x. You will find that x and y are the
same, and if you multiply them together and divide by 2 you will have
the maximum area.

I hope this helps. Write back if you have any questions, or if my
explanation needs more clarification. Good luck.

- Doctor Jaffee, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Calculus

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