Question

Bill's Coffee House, a store that specializes in coffee, has available 75 pounds (lb) of \(A\) grade coffee and \(120 \mathrm{lb}\) of \(B\) grade coffee. These will be blended into 1-lb packages as follows: an economy blend that contains 4 ounces (oz) of \(A\) grade coffee and 12 oz of \(B\) grade coffee, and a superior blend that contains 8 oz of \(A\) grade coffee and 8 oz of \(B\) grade coffee. (a) Using \(x\) to denote the number of packages of the economy blend and \(y\) to denote the number of packages of the superior blend, write a system of linear inequalities that describes the possible numbers of packages of each kind of blend. (b) Graph the system and label the corner points.

Short answer

For Grade A coffee: 4x + 8y ≤ 1200. For Grade B coffee: 12x + 8y ≤ 1920. Non-negativity constraints are x ≥ 0 and y ≥ 0.

Step-by-step solution

Define Variables

Let x be the number of economy blend packages and y be the number of superior blend packages.

Set Up the Inequality for Grade A Coffee

Each economy blend package uses 4 oz of Grade A coffee and each superior blend package uses 8 oz of Grade A coffee. Since there are 75 lbs of Grade A coffee available, which is equivalent to 1200 oz (because 1 lb = 16 oz), the inequality is: \[ 4x + 8y \leq 1200 \]

Set Up the Inequality for Grade B Coffee

Each economy blend package uses 12 oz of Grade B coffee and each superior blend package uses 8 oz of Grade B coffee. Since there are 120 lbs of Grade B coffee available, which is equivalent to 1920 oz, the inequality is: \[ 12x + 8y \leq 1920 \]

Set Up the Non-Negativity Constraints

The number of packages of each type can't be negative, so: \[ x \geq 0 \] \[ y \geq 0 \]

Graph the System of Linear Inequalities

To graph the system, first convert the inequalities to equalities to find the boundary lines: \[ 4x + 8y = 1200 \] \[ 12x + 8y = 1920 \] For each equation, find the x- and y-intercepts by setting the other variable to 0 and solving.

Find and Label the Corner Points

Solve the system of equations to find the intersection points. Check where the lines intersect the axes: - For \((4x + 8y = 1200)\) intercepts: x = 300, y = 150 - For \((12x + 8y = 1920)\) intercepts: x = 160, y = 240 Determine the feasible region from satisfying all constraints. The corner points can be found at the intercepts of the lines and where the lines intersect.

Key concepts

Systems of Inequalities

A system of inequalities is a set of two or more inequalities that are considered simultaneously. In the context of our coffee blend problem, we need to set up a system that takes into account both the amount of Grade A and Grade B coffee available. Let's break it down:
- The economy blend uses 4 ounces of Grade A coffee and 12 ounces of Grade B coffee per package.
- The superior blend uses 8 ounces of Grade A coffee and 8 ounces of Grade B coffee per package.
To formulate our system of inequalities:
1. Define variables: Let x be the number of economy blend packages and y be the number of superior blend packages.
2. Create inequalities for each type of coffee:
- For Grade A coffee: \[ 4x + 8y \leq 1200 \] (because there are 75 pounds, and 1 pound is 16 ounces, making it 1200 ounces)
- For Grade B coffee: \[ 12x + 8y \leq 1920 \] (because there are 120 pounds, making it 1920 ounces)
3. Add non-negativity constraints because we can't create a negative number of packages:
\[ x \geq 0 \]
\[ y \geq 0 \]
This system tells us all the possible combinations of economy and superior blend packages we can produce while staying within our coffee limits.

Graphing Linear Inequalities

Graphing linear inequalities involves shading the region of the coordinate plane where the solutions to the inequalities lie. Let's graph our system of inequalities for Bill's Coffee House:
1. Convert inequalities to equalities to find the boundary lines:
For Grade A coffee: \[ 4x + 8y = 1200 \]
For Grade B coffee: \[ 12x + 8y = 1920 \]
2. Find intercepts for each line:
- For \[4x + 8y = 1200\]:
If x = 0, y = 150 (Intercept on y-axis)
If y = 0, x = 300 (Intercept on x-axis)
- For \[12x + 8y = 1920\]:
If x = 0, y = 240 (Intercept on y-axis)
If y = 0, x = 160 (Intercept on x-axis)
3. Plot these boundary lines on a graph.
4. Identify the region satisfying the inequalities. This is done by checking a test point (commonly (0,0)) in each inequality to see if it satisfies the condition. If it does, shade towards that region; if not, shade the opposite side.
The overlapping shaded area is the feasible region. This is where all the constraints of the system are met simultaneously.

Corner Points

The corner points, also known as vertices, are crucial in systems of inequalities because they represent potential solutions that lie on the boundary of the feasible region. To find these corner points:
1. Identify intersections of boundary lines and non-negativity constraints.
2. Solve the system of equations to find these intersections:
- Intersection of \[4x + 8y = 1200\] and \[12x + 8y = 1920\]:
Solve for x and y simultaneously:
By substitution or elimination, we find that x = 120 and y = 105.
3. Include intercept points where boundary lines meet the axes:
For \[4x + 8y = 1200\], we have points (300,0) and (0,150)
For \[12x + 8y = 1920\], we have points (160,0) and (0,240)
4. Thus, our corner points are found at: (300,0), (0,150), (160,0), (0,240), and (120,105).
These points are key because they represent the extreme limits of our feasible solutions, helping us determine the best possible blend combinations within the given constraints.