Q. 59 - Precalculus Enhanced with Graphing Utilities

Question

Blending Coffee Bill’s Coffee House, a store that specializes in coffee, has available 75 pounds (lb) of A grade coffee and 120 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 number of packages of each kind of blend.

(b) Graph the system and label the corner points. 

Step-by-Step Solution

Answer

(a) $\left\{\begin{array}{l}x+2y\le 300\\ 3x+2\le 480\\ x\ge 0\\ y\ge 0\end{array}\right.$

(b) Corner points are $(0,0),(0,150),(90,105),(160,0)$

Part (a) Step 1. Given information

The goal of the task is to write a system of linear inequalities that describes the possible number of packages of each type of mixture x - the number of packages of the economy blend; y - the number of packages of the superior blend

Part (a) Step 2. Make a table

	A grade coffee (75)	B grade coffee (120)
x	4	12
y	8	8
	$16\left(75\right)=1200$	$16\left(120\right)=1920$

Part (a) Step 3. Find the system of linear inequalities

In the package of 1 kg we have an economical mixture and a top mixture, so it is: $x\ge 0 ; y\ge 0$.

$$\begin{gathered} 4x+8y \le 1200 \\ 12 x+8y \le 1920 \end{gathered}$$

Divide both the equations by 4,

$$\begin{gathered} x+2y \le 300 \\ 3x+2y \le 480 \end{gathered}$$

The system of linear inequalities is :

$\left\{\begin{array}{l}x+2y\le 300\\ 3x+2\le 480\\ x\ge 0\\ y\ge 0\end{array}\right.$

Part (b) Step 1. Given information

The goal of the problem is to graph the system of linear inequalities.

$\left\{\begin{array}{l}x+2y\le 300\\ 3x+2\le 480\\ x\ge 0\\ y\ge 0\end{array}\right.$

Part (b) Step 2. Graph the system

From the graph the corner points are :

$(0,0),(0,150),(90,105),(160,0)$