Problem 53 - Algebra and Trigonometry

Question

Coffee Blends \(A\) 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 lo of robusta beans available. Find a system of inequalities that describes the possible number of Standard and Deluxe packages the merchant can make. Graph the solution set.

Step-by-Step Solution

Answer

The inequalities are: \( 4x + 10y \leq 1280 \), \( 12x + 6y \leq 1440 \), \( x \geq 0 \), \( y \geq 0 \).

Step 1: Define the Variables

Let \( x \) be the number of Standard packages and \( y \) be the number of Deluxe packages produced by the coffee merchant. These variables will represent the number of each type of package that can be made.

Step 2: Formulate the Inequality for Arabica Beans

Each Standard package uses 4 oz of arabica, and each Deluxe package uses 10 oz. The merchant has 80 lb of arabica available, which is equivalent to \( 80 \times 16 = 1280 \) oz. Therefore, the inequality is:\[ 4x + 10y \leq 1280 \]

Step 3: Formulate the Inequality for Robusta Beans

Each Standard package uses 12 oz of robusta, and each Deluxe package uses 6 oz. The merchant has 90 lb of robusta available, which is equivalent to \( 90 \times 16 = 1440 \) oz. Therefore, the inequality is:\[ 12x + 6y \leq 1440 \]

Step 4: Include the Non-Negativity Constraints

Since the merchant cannot produce a negative number of packages, we have:\[ x \geq 0 \] \[ y \geq 0 \]

Step 5: Graph the Solution Set

To graph the solution set, first convert the inequalities into equalities to find the boundary lines: 1. For \( 4x + 10y = 1280 \): Find intercepts or other points to plot the line. - When \( x=0 \), \( y = 128 \) - When \( y=0 \), \( x = 320 \)1. For \( 12x + 6y = 1440 \): Find intercepts or other points to plot the line. - When \( x=0 \), \( y = 240 \) - When \( y=0 \), \( x = 120 \)Shade the region that satisfies all the inequalities, including the non-negativity constraints.

Key Concepts

Linear Programming

Linear programming is a powerful mathematical technique used to find the best outcome in a given search for maximum or minimum conditions, generally under a set of specified constraints. In our coffee blends problem, we are looking for the optimal combination of Standard and Deluxe coffee packages. This combination must satisfy certain restrictions, like limited resources. Linear programming requires expressing these conditions in the form of linear inequalities and then finding the feasible region where these inequalities overlap. The overall goal is to either maximize or minimize a function, often referred to as the 'objective function', which represents a particular measure of interest, such as profit or cost.

Graphing Inequalities

Graphing inequalities involves plotting lines on a coordinate plane that represent each constraint in a problem. These lines divide the plane into regions where different conditions hold true. For example, in our coffee blends problem, each inequality, such as \(4x + 10y \leq 1280\) and \(12x + 6y \leq 1440\), is a line that carves out boundaries within which possible solutions exist. To graph these inequalities, we first convert them into equalities to find the boundary lines. Then, using test points or intercepts, we determine which side of the line satisfies the inequality. The solution set is represented by the overlapping shaded region that satisfies all inequalities including non-negativity constraints. This is where feasible solutions can be found.

Non-Negativity Constraints

Non-negativity constraints are a fundamental part of linear programming that ensure practical solutions. They essentially state that negative quantities are not permissible in the context of a real-world problem, such as producing a negative number of coffee packages. For our coffee blends problem, these constraints are represented by \(x \geq 0\) and \(y \geq 0\). These inequalities ensure that the solution remains within a realistic frame, preventing the merchant from expecting negative amounts of Standard or Deluxe coffee blends. These constraints often help bound the solution space within the first quadrant of a graph, as only non-negative values for our variables make sense in this scenario.

Coffee Blends Problem

The coffee blends problem is an example of how linear programming is used in real-world applications. The merchant must decide how many Standard and Deluxe packages to create, using limited amounts of arabica and robusta beans.

• The Standard blend uses up more robusta, while the Deluxe blend requires more arabica. This creates a trade-off that must be managed to satisfy supply restrictions.
• By setting up inequalities based on the resources available and the requirements of each blend, the merchant can find a range of feasible production scenarios.
• These feasible scenarios are represented graphically, showing possible solutions that can economically satisfy demand and resource constraints.

Understanding and solving these kinds of problems allows businesses to efficiently allocate their resources while meeting both supply limits and market demands. Through this, manufacturers can optimize their outputs, ensuring the best possible combination of products given the constraints they face.