A timber rectangular beam is to be designed where the objective function

is such that $W=(x_1{x}_2+0\mathrm{.}2x_2^2)l,$ Where $x_1$ and $x_2$ are the

width and depth of the beam. $$ is the weight per unit length $l.$ The

following constraints are applicable.

flexural stress $=\frac{270}{x_1{x}_2^2}1\mathrm{.}0$

Deflection $limit=\frac{2020}{x_1{x}_2^3}1\mathrm{.}0$

Using the method of Lagrange multipliers, calculate the values of $x_1$

and $x_2$ and $w$ for the optimum design. $($ Please solve it in handwriting and in$$detail$)$