If the difference between the maximum and minimum values of the objective function z =7 x-8 y subject to the constraints x+y 20 , y 5, x, y 0 is 5 k+200 , then the value of k is

Step-by-step solution

Identifying feasible region vertices The feasible region is bounded by the constraints: x + y 20 , y 5 , and x 0 . Intersection of x = 0 and y = 5 gives vertex (0,5) . Intersection of y = 5 and x + y = 20 gives vertex (15,5) . Intersection of x = 0 and x + y = 20 gives vertex (0,20) . Evaluating objective function z = 7x - 8y at (0,5) yields z = -40 , at (15,5) yields z = 65 , and at (0,20) yields z = -160 . The maximum is 65 and the minimum is -160 . Solving for k Maximum - Minimum = 65 - (-160) = 225 225 = 5k + 2