Q. 60 - Precalculus Enhanced with Graphing Utilities

Question

Mixed Nuts Nola’s Nuts, a store that specializes in selling nuts, has available 90 pounds (lb) of cashews and 120 lb of peanuts. These are to be mixed in 12-ounce (oz) packages as follows: a lower-priced package containing 8 oz of peanuts and 4 oz of cashews and a quality package containing 6 oz of peanuts and 6 oz of cashews. 

(a) Use x to denote the number of lower-priced packages and use y to denote the number of quality packages. Write a system of linear inequalities that describes the possible number of each kind of package. (b) Graph the system and label the corner points 

Step-by-Step Solution

Answer

$$\begin{gathered} \left(a\right) \left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ 2x+3y\le 540\\ 4x+3y\le 720\end{array}\right. \\ \left(b\right) \mathrm{Corner} \mathrm{points} \mathrm{are} (0,0),(0,180) \mathrm{and} (135,0). \end{gathered}$$

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 each type of packet x - the number of lower-priced packages; y - the number of quality packages

Part (a) Step 2. Make a table

	Cashews (90)	Peanuts (120)
x	4	8
y	6	6

$$\begin{gathered} 12\left(90\right)=1080 \\ 12\left(120\right)=1440 \end{gathered}$$

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

We get a system of inequalities

$$\begin{gathered} 4x+6y\le 1080 \\ 8x+6y\le 1440 \\ \mathrm{Divide} \mathrm{both} \mathrm{the} \mathrm{equations} \mathrm{by} 2, \\ 2\mathrm{x}+3\mathrm{y}\le 540 \\ 4\mathrm{x}+3\mathrm{y}\le 720 \\ \mathrm{The} \mathrm{system} \mathrm{of} \mathrm{linear} \mathrm{inequalities} \mathrm{is} : \\ \left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ 2x+3y\le 540\\ 4x+3y\le 720\end{array}\right. \end{gathered}$$

Part (b) Step 1. Given information

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

$\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ 2x+3y\le 540\\ 4x+3y\le 720\end{array}\right.$

Part (b) Step 2. Graph the system