Q. 58 - Precalculus Enhanced with Graphing Utilities

Question

A store that specializes in selling nuts has available $72$ 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

(a) The system of linear inequalities that describes the possible number of each kind of package is $\left\{\begin{array}{l}0.5x+0.375y\le 120\\ 0.25x+0.375y\le 72\end{array}\right.$

(b) The graph of the system of inequality is:

Part (a) Step 1. Given

Total amount of cashews is $72$ pounds and peanuts is $120$ pounds.

Lower priced package has $8 oz$ of peanuts and $4 oz$ of cashews.

A quality package contains $6 oz$ of peanuts and $6 oz$ of cashews.

Step 2. Format the conditions.

For peanuts,

The total amount of peanuts is $120$ pounds.

A lower priced package has $12$ ounces, so,

Convert ounces into pounds: $12 oz=0.75 lb$

So the lower priced package has $0.5 lb$ of peanuts and $0.25 lb$ of cashews.

The quality package contains $0.375 lb$ of both peanuts and cashews.

Part (a) Step 3. Form system of inequality.

The above condition can be applied to inequality such that, \

Amount of peanuts$=0.5x+0.375y\le 120$

Amount of cashews $=0.25x+0.375y\le 72$

Part (b) Step 1. To graph the system and label the corner points.

Graph the system of inequalities.