How can we solve this geometry problem using Lagrange Multipliers? It is must to use the given formula of area.

Question

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The problem is belongs to Mathematical Modeling with Excel book.

Answer 1

With $A = \frac 12 a b\sin\theta$, the lagrangian can be build as

$$ L(a,b,c,\theta,\lambda_1,\lambda_2) = A + \lambda_1(a+b+c-l_0)+\lambda_2(c^2-a^2-b^2+2a b\cos\theta) $$

after that, the stationary points determination is made by solving

$$ \nabla L = 0 = \left\{ \begin{array}{l} \lambda_2 (2 b \cos (\theta )-2 a)+\frac{1}{2} b \sin (\theta )+\lambda_1 \\ \lambda_2 (2 a \cos (\theta )-2 b)+\frac{1}{2} a \sin (\theta )+\lambda_1\\ 2 c \lambda_1+\lambda_2\\ \frac{1}{2} a b \cos (\theta )-2 a b \lambda_2 \sin (\theta ) \\ a+b+c-l_0 \\ 2 a b \cos (\theta )-a^2-b^2+c^2 \\ \end{array} \right. $$

giving

$$ \left( \begin{array}{ccccccc} A & a & b & c & \theta& \lambda_1 & \lambda_2\\ 0 & 0 & \frac{l_0}{2} & \frac{l_0}{2} & 0 & 0 & 0 \\ 0 & 0 & \frac{l_0}{2} & \frac{l_0}{2} & \pi & 0 & 0 \\ 0 & \frac{l_0}{2} & 0 & \frac{l_0}{2} & 0 & 0 & 0 \\ 0 & \frac{l_0}{2} & 0 & \frac{l_0}{2} & \pi & 0 & 0 \\ -\frac{l_0^2}{12 \sqrt{3}} & \frac{l_0}{3} & \frac{l_0}{3} & \frac{l_0}{3} & -\frac{\pi }{3} & \frac{l_0}{6 \sqrt{3}} & -\frac{1}{4 \sqrt{3}} \\ \frac{l_0^2}{12 \sqrt{3}} & \frac{l_0}{3} & \frac{l_0}{3} & \frac{l_0}{3} & \frac{\pi }{3} & -\frac{l_0}{6 \sqrt{3}} & \frac{1}{4 \sqrt{3}} \\ \end{array} \right) $$