Question

Manufacturing Trucks Mike's Toy Truck Company manufactures two models of toy trucks, a standard model and a deluxe model. Each standard model requires 2 hours (h) for painting and \(3 \mathrm{~h}\) for detail work; each deluxe model requires \(3 \mathrm{~h}\) for painting and \(4 \mathrm{~h}\) for detail work. Two painters and three detail workers are employed by the company, and each works \(40 \mathrm{~h}\) per week. (a) Using \(x\) to denote the number of standard-model trucks and \(y\) to denote the number of deluxe-model trucks, write a system of linear inequalities that describes the possible numbers of each model of truck that can be manufactured in a week. (b) Graph the system and label the corner points.

Short answer

The system of inequalities is: \[ \begin{cases} 2x + 3y \leq 80 \ 3x + 4y \leq 120 \ x \geq 0 \ y \geq 0 \end{cases} \]

Step-by-step solution

Define Variables

Let \(x\) be the number of standard model toy trucks and \(y\) be the number of deluxe model toy trucks.

Formulate Painting Time Constraint

Each standard model requires 2 hours and each deluxe model requires 3 hours for painting. The total hours available for painting is 2 painters \times 40\ hours = 80 hours. Therefore, the inequality is: \(2x + 3y \leq 80\).

Formulate Detail Work Time Constraint

Each standard model requires 3 hours and each deluxe model requires 4 hours for detail work. The total hours available for detail work is 3 detail workers \times 40\ hours = 120 hours. Therefore, the inequality is: \(3x + 4y \leq 120\).

Non-Negativity Constraints

The number of toy trucks manufactured cannot be negative, so the inequalities are \(x \geq 0\) and \(y \geq 0\).

System of Inequalities

Combining all the inequalities, the system is: \[ \begin{cases} 2x + 3y \leq 80 \ 3x + 4y \leq 120 \ x \geq 0 \ y \geq 0 \end{cases} \]

Graph the System

Graph the inequalities on the Cartesian plane. Use the intercepts and test points to determine the feasible region. Label the intersection points of the boundary lines as corner points.

Find and Label the Corner Points

Solve for intersection points of the lines \(2x + 3y = 80\) and \(3x + 4y = 120\). Also, consider intercepts of each line with the axes. These corner points typically define the feasible region.

Key concepts

System of Linear Inequalities

In linear programming, a system of linear inequalities represents the conditions or constraints we need to consider. It's like a set of rules that limits what choices we can make. In our example, the system of linear inequalities tells us how many toy trucks can be painted and detailed in a week based on the time each task takes.

For Mike's Toy Truck Company:

• We have two inequalities related to painting and detail work.
• Then, additional non-negativity constraints ensure we can't produce a negative number of trucks.

Combining these gives us a system that looks like this: \[ \begin{cases} 2x + 3y \ \leq 80 \ 3x + 4y \ \leq 120 \ x \ \geq 0 \ y \ \geq 0 \ \ \end{cases} \]

Feasible Region

The feasible region is the area on the graph where all the inequalities overlap. This region contains all the possible solutions that satisfy the constraints. Essentially, it's the 'zone' where we can produce toy trucks without violating any time limits for painting and detail work.

To find this region, you:

• First, graph each inequality on the Cartesian plane.
• The area where all shaded regions intersect is the feasible region.

The feasible region will be a polygon, and we will focus on this area to find the possible combinations of standard and deluxe toy trucks that we can manufacture.

Constraints

Constraints are the limitations or restrictions given in the problem. In Mike's Toy Truck Company example, the constraints are:

• The painting time for standard and deluxe models, restricted by the painters' total available hours.
• The detailing time for standard and deluxe models, restricted by the detail workers' total available hours.
• Non-negativity constraints, ensuring that production numbers cannot be negative.

So we have: \[ \begin{cases} 2x + 3y \ \leq 80 \ 3x + 4y \ \leq 120 \ x \ \geq 0 \ y \ \geq 0 \ \end{cases} \]
Understanding the constraints helps us set up the feasible region and find the best solution.

Corner Points

Corner points are the vertices of the feasible region polygon. These points are crucial because they represent the possible optimal solutions to our problem.

Here's how to find and use them:

• Identify where the boundary lines intersect with each other and the axes.
• In our toy truck example, calculate where the lines \(2x + 3y = 80\) and \(3x + 4y = 120\) intersect.
• Also, find where each line intersects the x and y axes.

These points will give us different combinations of toy trucks, which we can evaluate to find the most efficient production mix.

Graphing Inequalities

Graphing inequalities is the key to visually understanding the feasible region. You start by plotting each inequality on a graph. Each inequality creates a half-plane where solutions are valid.

Here's how to graph inequalities:

• For each inequality, first convert it to an equation (replace ≤ or ≥ with =).
• Plot the line by finding the intercepts.
• Shade the region that satisfies the inequality (use test points if uncertain).

By doing this for each constraint, you'll see the overlapping shaded area, which is your feasible region. This visual representation helps immensely in understanding the problem and finding the corner points.