SOLUTION: An isosceles triangle has perimeter 64m. Find the dimensions of this triangle that make its area a maximum

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Question 450607: An isosceles triangle has perimeter 64m. Find the dimensions of this triangle that make its area a maximum
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
Suppose the side lengths are x, x, and 64-2x:


(64-2x won't show up so assume the base length is 64-2x)

By Heron's formula, we can find the area A of the triangle in terms of x (semiperimeter = 32)



Maximizing the area requires a little calculus. We can say that A is a function in terms of x, and find dA/dx. However, we can also note that the value of x that maximizes A will also be the value of x that maximizes A^2. To make things simpler, we can find A^2:



Here, we take the derivative with respect to x:





Here, the derivative is equal to zero when x = 32 or x = 64/3. Clearly, x = 64/3 maximizes the area of the triangle. This would also imply that the maximal area occurs when the triangle is equilateral.