Main Answer:
The optimal solution for the linear programming problem, with the objective function \(g = x + 2y\) and the given constraints \(8x + y \geq 85\), \(x + y \geq 50\), \(x + 4y \geq 80\), and \(x + 10y \geq 104\), subject to \(x \geq 0\) and \(y \geq 0\), is at the point (10, 40), with the minimum value of \(g\) being 90.
Step-by-step explanation:
To solve the linear programming problem graphically, we first plot the feasible region determined by the system of inequalities. The feasible region is the intersection of the shaded regions satisfying all the constraints. For this problem, the feasible region is a polygon bounded by the lines \(8x + y = 85\), \(x + y = 50\), \(x + 4y = 80\), and \(x + 10y = 104\), along with the non-negativity constraints.
Next, we evaluate the objective function \(g = x + 2y\) at the corner points of the feasible region. The point (10, 40) yields the minimum value of \(g\) at 90. This indicates that to minimize \(g\) within the given constraints, the optimal values for \(x\) and \(y\) are 10 and 40, respectively.
Understanding the graphical method in linear programming allows for effective visualization of solutions and aids in making informed decisions based on the objective function and constraints.