Problem 10 - College Algebra and Calculus: An Applied Approach

Question

Sketch the region determined by the constraints. Then find the minimum anc maximum values of the objective function and where they occur, subject to the indicated constraints. Objective function: $$ z=7 x+8 y $$ Constraints: $$ \begin{aligned} x & \geq 0 \\ y & \geq 0 \\ x+2 y & \leq 8 \end{aligned} $$

Step-by-Step Solution

Answer

The objective function has a minimum value of 0 at the point (0,0) and a maximum value of 56 at the point (8,0).

Step 1: Sketch the Constraints

The constraints given are \(x \geq 0\), \(y \geq 0\), and \(x + 2y \leq 8\). To sketch them, start by sketching the lines \(x = 0\), \(y = 0\), and \(x + 2y = 8\) in the same xy-plane. The line \(x = 0\) is the y-axis and the line \(y = 0\) is the x-axis. The line \(x + 2y = 8\) intersects the x-axis at (8,0) and the y-axis at (0,4). Connect these two points with a straight line.

Step 2: Identify the Feasible Region

The feasible region is the solution set of the system of inequalities. It can be found by shading the region that meets all of the constraints. Here, it is the region above the x-axis (due to \(y \geq 0\)), to the right of the y-axis (due to \(x \geq 0\)), and beneath the line \(x + 2y = 8\) (due to \(x + 2y \leq 8\)). The feasible region is thus the triangle with vertices at the origin (0,0), the x-intercept of the line \(x + 2y = 8\), (8,0), and the y-intercept of the same line, (0,4).

Step 3: Evaluate at Vertices

To find the minimum and maximum values of the objective function \(z = 7x + 8y\) over the feasible region, plug the coordinates of each vertex into the objective function. For the vertex (0,0), \(z = 7(0) + 8(0) = 0\). For the vertex (8,0), \(z = 7(8) + 8(0) = 56\). For the vertex (0,4), \(z = 7(0) + 8(4) = 32\).

Step 4: Determine Minima and Maxima

The smallest value of the objective function over the feasible region is the minimum and the largest value is the maximum. Here, the minimum value of the objective function is 0 at (0,0), and the maximum value is 56 at (8,0).

Key Concepts

Objective Function

In linear programming, the objective function is the formula you want to optimize, such as maximizing profits or minimizing costs. This function is expressed in terms of variables, usually representing real-world quantities, like products or resources. In the given exercise, the objective function is \( z = 7x + 8y \). Here, \( x \) and \( y \) represent variables, while 7 and 8 are coefficients that weigh these variables in the context of optimization. To maximize or minimize the objective function, we evaluate its value at various points within the feasible region, which are determined by intersections of the constraints. The goal is to find which values of \( x \) and \( y \) yield the best outcome for \( z \). By understanding how each variable affects \( z \), you can make informed decisions about resource allocation.

Constraints

Constraints in linear programming are the conditions or limits placed on potential solutions. These are expressed as inequalities and define the boundaries of what's feasible or possible. In our exercise, the constraints are:

• \( x \geq 0 \)
• \( y \geq 0 \)
• \( x + 2y \leq 8 \)

These inequalities represent conditions such as resource limitations or production capacities.Breaking down these constraints:
- \( x \geq 0 \) and \( y \geq 0 \) ensure that we only consider the first quadrant of the coordinate plane where both values are non-negative. This corresponds to the real-world scenarios where negative quantities (like products or hours) are nonsensical.- \( x + 2y \leq 8 \) imposes a limit on the combined values of \( x \) and \( y \). It sets a boundary and shapes the feasible region.

Feasible Region

The feasible region is the graphical representation of all possible solutions that satisfy the constraints. It is often a polygonal area on the coordinate plane and represents the domain where the objective function can be evaluated.In the exercise, the feasible region is formed by:

• The x-axis where \( y \) is non-negative, \( y \geq 0 \)
• The y-axis where \( x \) is non-negative, \( x \geq 0 \)
• The line \( x + 2y = 8 \), dictating the upper boundary \( x + 2y \leq 8 \)

The region where these conditions overlap is a triangle with vertices at (0,0), (8,0), and (0,4). Inside this triangle is the solution space for the objective function.
Evaluating the objective function within this area ensures that the solutions are valid and adhere to all constraints.

Vertices

Vertices are the corner points of the feasible region. These are crucial in linear programming because the optimum values of the objective function (either maximum or minimum) will occur at one of these vertices or along the edges.In the given problem, the vertices are located at:

• (0,0)
• (8,0)
• (0,4)

To find the best solution for the objective function, evaluate it at each vertex. This is because the linear nature of both the constraints and the objective function guarantees that extrema lie on vertices.- At (0,0): \( z = 0 \)
- At (8,0): \( z = 56 \)
- At (0,4): \( z = 32 \)
The maximum value of \( z \), 56, occurs at the vertex (8,0), while the minimum value, 0, is at (0,0). Evaluating at vertices is an efficient and surefire way to pinpoint optimal results in linear programming.