In Exercises 1–4, find the value of the objective function at each corner of the graphed region. What is the maximum value of the objective function? What is the minimum value of the objective function? 1. Objective Function z=40x+50y

Verified step by step guidance

1. Identify the corner points of the graphed region: (0,0), (0,10), (6,8), and (10,0).
2. Substitute each corner point into the objective function z = 40x + 50y to find the value of z at each point.
3. Calculate z for the point (0,0): z = 40(0) + 50(0).
4. Calculate z for the point (0,10): z = 40(0) + 50(10).
5. Calculate z for the point (6,8): z = 40(6) + 50(8).

Key Concepts

Objective Function

An objective function is a mathematical expression that defines the goal of an optimization problem, typically in the form of maximizing or minimizing a value. In this case, the objective function is z = 40x + 50y, where x and y are variables representing quantities to be optimized. The values of the objective function are evaluated at specific points, known as corner points, within the feasible region.

Feasible Region

The feasible region is the set of all possible points that satisfy the constraints of an optimization problem. It is typically represented graphically as a polygon on a coordinate plane. In this scenario, the feasible region is bounded by the axes and the lines representing the constraints, and the corner points of this region are critical for evaluating the objective function.

Corner Points

Corner points, or vertices, of the feasible region are the points where the boundary lines intersect. These points are essential in linear programming because the maximum and minimum values of the objective function will occur at one of these corner points. In this problem, the corner points are (0, 10), (6, 8), and (10, 0), and evaluating the objective function at these points will yield the required maximum and minimum values.