Consider the following resource allocation problem:

maxz$=5x_1+4x_2+3x_3$

subject $to$:

$x_1+x_{3}15,(resource1)$

$x_2+2x_{3}25,(resource2)$

$x_1,x_2,x_{3}0$

$($a$)$ Construct the dual problem for this primal problem

$($b$)$ Solve the dual problem graphically

$($c$)$ From the graphical solution obtained in $($b$),$ find the shadow prices for $($resource $1)$ and $($resource $2)$ and the reduced costs for activities $x_1,x_2$ and $x_3.$

$2.$ By inspection only, find the optimal value of the following LP from its dual.

Minimize $z=10x_1+4x_2+6x_3$

subject $to5x_1-7x_2+2x_{3}25$

$x_1,x_2,x_{3}0$

Solve the following problem using Lagrange optimization and estimate the change in the optimal objective function value when the right hand side increases by $5\%,$ i$.$e$.$ the right hand side increases from $8$ to $8\mathrm{.}4.$

max$2x+y$

subject $to4x^2+y^2=8$

You want to minimize the surface area of a cone$-$shaped drinking cup having fixed volume $V_0.$ Solve the problem as a constrained optimization problem. To simplify the algebra, minimize the square of the area. The area is $r\sqrt[\phantom{2}]{r^2+h^2}.$ Solve the problem using Lagrange multipliers. Hint. You can assume that $r0$ in the optimal solution.