Q: Area of a triangle with sides of length $x,$ $y,$ and $z$ is $\sqrt{a(a-x)(a-y)(a-z)},$ where $$ is half the perimeter of the triangle.
There is 100 m. cable to make a triangular shaped area.
a) Formulate a NLP to maximize the area of the triangular area.
b) Is it possible to solve the formulated NLP using Lagrange multipliers method? Why?
c) If your answer is YES in part (b), solve the NLP with Lagrange multipliers methods. Otherwise solve it using an appropriate method.
So i ve just reached this equations:
$$ \max z = \sqrt{( − )( − )( − )} $$
$$ \text{s.t. } \begin{align} & x + y + z = 100 \\ & a = (x + y + z)/2 \\ & a\le b + c, b\le a + c, c\le a + b \\ & a\ge 0, b\ge 0, c\ge 0 \end{align} $$
But cant go further, I'm stuck on Lagrange, so not sure how to do Qb and Qc?
You can square the area,as it's easier to work with. So your optimization is $r=a(a-x)(a-y)(a-z)$ with constraint $x+y+z=100$ and $a=(x+y+z)/2$. You're better off first simplifying this. For example $a=200$ follows from the two constraints, so you may as well treat $a$ as constant.
The derivative in $x$ of r is $-a(a-y)(a-z)$ and $1$ for the constraint, so with Lagrange multiplier $L$ this gives you:
$$a(a-y)(a-z)=a(a-x)(a-z)=a(a-y)(a-z)=L.$$
This implies $x=y=z=100/3$. You can verify further that this indeed a maximum. Not surprisingly the equilateral triangle maximizes the area.