Question

A small truck can carry no more than 1600 pounds (Ib) of cargo and no more than 150 cubic feet \(\left(\mathrm{ft}^{3}\right)\) of cargo. A printer weighs \(20 \mathrm{lb}\) and occupies \(3 \mathrm{ft}^{3}\) of space. A microwave oven weighs 30 lb and occupies \(2 \mathrm{ft}^{3}\) of space. (a) Using \(x\) to represent the number of microwave ovens and \(y\) to represent the number of printers, write a system of linear inequalities that describes the number of ovens and printers that can be hauled by the truck. (b) Graph the system and label the corner points.

Short answer

The system of inequalities is:\[ 30x + 20y \leq 1600 \]\[ 2x + 3y \leq 150 \]\[ x \geq 0 \]\[ y \geq 0 \].

Step-by-step solution

Define the variables

Let \( x \) represent the number of microwave ovens and \( y \) represent the number of printers.

Write the weight inequality

Each microwave oven weighs 30 pounds and each printer weighs 20 pounds. The total weight constraint is given by the inequality:\[ 30x + 20y \leq 1600 \]

Write the volume inequality

Each microwave oven occupies 2 cubic feet and each printer occupies 3 cubic feet. The total volume constraint is given by the inequality:\[ 2x + 3y \leq 150 \]

Write the non-negativity constraints

Since the number of microwave ovens and printers cannot be negative, we also include the inequalities:\[ x \geq 0 \]\[ y \geq 0 \]

Graph the system of inequalities

Plot the inequalities:- Convert \( 30x + 20y \leq 1600 \) to its boundary line \( 30x + 20y = 1600 \) and plot it. The area of interest is below or on this line.- Convert \( 2x + 3y \leq 150 \) to its boundary line \( 2x + 3y = 150 \) and plot it. The area of interest is below or on this line.- Include the non-negativity constraints by considering only the first quadrant (where \( x \geq 0 \) and \( y \geq 0 \)).

Find and label the corner points

Identify the points where the boundary lines intersect each other and the axes. These are the corner points of the feasible region. Calculate the intersection points algebraically:1. Intersection of \( 30x + 20y = 1600 \) and \( 2x + 3y = 150 \).2. Intersection of each line with the axes (\( y = 0 \) and \( x = 0 \)).Solve these systems of equations to find the exact coordinates of the corner points.

Key concepts

Linear Inequality

A linear inequality is similar to a linear equation but with an inequality sign instead of an equals sign. Inequalities can be represented using symbols such as \( \leq \, \geq \, < \, \( \) > \). In our example, we have two linear inequalities:

• \( 30x + 20y \leq 1600 \) represents the weight constraint.
• \( 2x + 3y \leq 150 \) represents the volume constraint.

These inequalities imply that the total weight and the total volume cannot exceed the truck's carrying capacity. To set up these inequalities, identify the constraints, use appropriate coefficients for each variable, and ensure the inequality correctly reflects the limitation.

Graphing Inequalities

Graphing inequalities involves plotting the boundary lines and shading the region that meets the inequality conditions. Start by converting the given inequalities into equations. For instance, convert \( 30x + 20y \leq 1600 \) into \( 30x + 20y = 1600 \). Plot these lines on a graph. The lines will intersect, creating a region.
To determine which side of the line to shade, choose a test point (e.g., (0,0)) and see if it satisfies the inequality. If it does, shade the region that includes that point. Otherwise, shade the opposite side. Also, include non-negativity constraints by plotting in the first quadrant (\(x \geq 0 \) and \( y \geq 0 \)).

Feasible Region

The feasible region is the solution set where all inequalities overlap on the graph. It's the area where the constraints of the problem are satisfied simultaneously. To find the feasible region, follow these steps:

• Plot the boundary lines of the inequalities.
• Determine the region that satisfies all inequalities. This is the shaded area where all individual shaded regions overlap.
• The vertices of this region are called corner points.

These corner points can be found by solving the system of equations at the boundaries' intersection points. In our case, you can find them by solving:

• \( 30x + 20y = 1600 \) and \( 2x + 3y = 150 \)
• The intersection points with the axes, i.e., when \( x = 0 \) and \( y = 0 \).

Each corner point represents a potential solution to the problem and helps determine the maximum number of microwaves and printers the truck can carry.