Problem 47 - Precalculus: Mathematical for Calculus

Question

A coffee merchant sells two different coffee blends. The Standard blend uses 4 oz of arabica and 12 oz of robusta beans per package; the Deluxe blend uses 10 oz of arabica and 6 oz of robusta beans per package. The merchant has 80 lb of arabica and 90 lb of robusta beans available. Find a system of inequalities that describes the possible number of Standard and Deluxe packages he can make. Graph the solution set.

Step-by-Step Solution

Answer

System: \(4x + 10y \leq 1280\), \(12x + 6y \leq 1440\), \(x \geq 0\), \(y \geq 0\). Graph shows feasible region.

Step 1: Understand the Variables

Let us define the variables for clarity. Let \( x \) represent the number of Standard packages and \( y \) represent the number of Deluxe packages. Each Standard package requires 4 oz of arabica and 12 oz of robusta. Each Deluxe package requires 10 oz of arabica and 6 oz of robusta.

Step 2: Set Up Inequalities for Arabica Beans

The merchant has a total of 80 lb (which is 80 * 16 oz = 1280 oz) of arabica beans. The inequality for arabica beans is: \(4x + 10y \leq 1280\).

Step 3: Set Up Inequalities for Robusta Beans

The merchant also has 90 lb (which is 90 * 16 oz = 1440 oz) of robusta beans. The inequality for robusta beans is: \(12x + 6y \leq 1440\).

Step 4: Non-negativity Constraints

Since the number of packages cannot be negative, we have the constraints: \(x \geq 0\) and \(y \geq 0\).

Step 5: Graph the System of Inequalities

To graph the system, plot the lines represented by the equalities of the inequalities on a coordinate plane: \(4x + 10y = 1280\) and \(12x + 6y = 1440\). Shade the region that satisfies all the inequalities, which includes the area below or on the lines (considering the non-negativity constraints as well).

Key Concepts

System of Inequalities

In mathematics, a system of inequalities is a set of two or more inequalities that are all considered simultaneously. In our example, the coffee merchant needs to solve a system of inequalities to determine how many Standard and Deluxe packages can be created without exceeding his bean resources.

The two primary inequalities are based on the available arabica and robusta beans:

• For the arabica beans: \( 4x + 10y \leq 1280 \)
• For the robusta beans: \( 12x + 6y \leq 1440 \)

These inequalities need to be solved together, meaning both have to be true at the same time. Each inequality describes a condition that the number of coffee packages must satisfy.

In any system of inequalities, it's essential to identify and use all available resources wisely to avoid exceeding supplies while meeting production needs.

Graphical Solution

A graphical solution provides a visual means to solve the system of inequalities. By plotting the inequalities on a coordinate plane, you can easily see the region where all conditions are met simultaneously.

To graph the system:

• First, convert each inequality into an equation by replacing the inequality signs with equal signs: \(4x + 10y = 1280\) and \(12x + 6y = 1440\).
• These become lines on the graph, which you plot by finding points that satisfy each equation.
• Shade below or on the lines to represent solutions to the inequalities. This shaded area is the feasible region.

The feasible region is where both conditions overlap, and within this area, all points represent possible solutions that do not exceed the limits of the merchant's bean supply.

Graphing is a powerful tool because it allows you to easily identify the best solution visually.

Non-negativity Constraints

Non-negativity constraints are vital in ensuring that certain variables, like the number of coffee packages, are not negative. In real-world scenarios, the number of items usually cannot be less than zero, which is why we apply these constraints.

For this coffee merchant:

• The constraint \( x \geq 0 \) ensures that the number of Standard packages is never negative.
• Similarly, \( y \geq 0 \) ensures that the number of Deluxe packages is also non-negative.

These constraints shape our feasible region on the graph to lie within the first quadrant where both x and y are non-negative.

Understanding and applying non-negativity constraints is crucial, as they reflect the practical side of many problems where resources and products need to be realistically accounted for.