Chapter 11.7, Problem 59AYU

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 numbers of packages of each kind of blend.

b. Graph the system and label the corner points.

To determine

To find:

1. To write a system of linear inequalities that describes the possible numbers of packages of each kind of blend.

2. To graph the system and label the corner points.

Answer to Problem 59AYU

Solution:

$x \ge 0, y \ge 0, x + 2y \le 300, 3x + 2y \le 480$

Explanation of Solution

Formula Used:

$1 oz = lb \times 16$

Given:

	A–grade	B–grade
Economy blend	4 oz	12 oz
Superior blend	8 oz	8 oz

Store has $75lb$ of A–grade coffee and $120lb$ of B–grade coffee.

Calculation:

Begin by assigning symbols for the two variables.

$x = \text{Number of packages of the economy blend}.$

$y = \text{Number of packages of the superior blend}.$

Because $x$ and $y$ represents the number of packages, the only meaningful values of $x$ and $y$ are non–negative.

Therefore, $x \ge 0, y \ge 0$

From the given data and the formula used we get

$4x + 8y \le 1200 => x + 2y \le 300$

$12x + 8y \le 1920 => 3x + 2y \le 480$

Therefore, a system of linear inequalities that describes the possible amounts of each investment are

$x \ge 0$

$y \ge 0$

$x + 2y \le 300$

$3x + 2y \le 480$

2. The graph of the system and corner points are illustrated in the figure below.