Agraph represents the feasible region for the system: y is less than or equal to 2x, x + y is greater than or equal to 45, and x is less than or equal to 30. what is the minimal number for the objective function p = 20x + 16y? what is the value of x then? what is the value at y then?

We are given a system of inequalities as:

y is less than or equal to 2x

i.e.          y ≤ 2x--------(1)

x + y is greater than or equal to 45

i.e.      x+y ≥ 45  ------------(2)    

and    x is less than or equal to 30.    

i.e.               x ≤ 30 -----------(3)    

On plotting these inequalities we get the boundary points as:

         (15,30) , (30,60) and (30,15)    

( Since, the optimal solution always exist at the boundary point )

The optimal function is given by:

 Minimize  P = 20x+16y

Hence, at (15,30) we get:

P= 780

at (30,60) we get:

  P= 1560

at (30,15) we get:

P= 840

This  means that the minimal value of the function is 780

and the value exist at (15,30)