Final answer:
The area of an isosceles triangle with a given perimeter p is maximized when the triangle is equilateral, meaning that the legs and base are all equal. Thus, the length of the legs to maximize the area should each be p/3.
Step-by-step explanation:
When optimizing the area of an isosceles triangle with a given perimeter p, one important consideration is that its area is maximum when the triangle is equilateral.
To understand why, we can use the concept of the triangle's base and height. The area of a triangle is given by 1/2 × base × height. In an isosceles triangle, the legs are equal, and if we denote the length of each leg as a, and the base as b, we can express the height h in terms of a using the Pythagorean theorem.
Since the perimeter p is equal to 2a + b, and we want to maximize the area, we need to solve for a and b in terms of p. It turns out that the area is maximized when the triangle is equilateral, which means all sides are equal, and the lengths are each p/3. Thus, to maximize the area of an isosceles triangle given its perimeter p, the legs (equal sides) should each be p/3.