Question

the california almond growers have at most 2800 boxes of almonds to be shipped from their plant in sacramento to des moines and san antonio. the des moines market needs at least 1200 boxes, while the san antonio market must have at least 600 boxes. let x = the number of boxes to be shipped to des moines and y = the number of boxes to be shipped to san antonio. complete parts (a) and (b). (a) write a system of inequalities to express the conditions of the problem. (b) graph the feasible region of the system. use the graphing tool to graph the system.

Explanation:

Step1: Analyze non - negative constraints

Since \(x\) and \(y\) represent the number of boxes, they cannot be negative. So we have \(x\geq0\) and \(y\geq0\).

Step2: Analyze the total shipment constraint

The total number of boxes to be shipped (\(x + y\)) is at most 2800. So \(x + y\leq2800\).

Step3: Analyze the Des Moines market constraint

The Des Moines market needs at least 1200 boxes, so \(x\geq1200\).

Step4: Analyze the San Antonio market constraint

The San Antonio market must have at least 600 boxes, so \(y\geq600\).

So the system of inequalities is:
\(

$$\begin{cases}x\geq0\\y\geq0\\x + y\leq2800\\x\geq1200\\y\geq600\end{cases}$$

\)

For part (b), to graph the feasible region:

• Graph the line \(x = 0\) (the y - axis) and shade the region to the right (since \(x\geq0\)).
• Graph the line \(y = 0\) (the x - axis) and shade the region above (since \(y\geq0\)).
• Graph the line \(x + y=2800\) (rewrite as \(y=-x + 2800\)). The inequality \(x + y\leq2800\) means we shade below this line.
• Graph the line \(x = 1200\) (a vertical line) and shade the region to the right (since \(x\geq1200\)).
• Graph the line \(y = 600\) (a horizontal line) and shade the region above (since \(y\geq600\)).

The feasible region is the intersection of all these shaded regions.

Answer:

(a) The system of inequalities is \(\boldsymbol{

$$\begin{cases}x\geq0\\y\geq0\\x + y\leq2800\\x\geq1200\\y\geq600\end{cases}$$

}\)
(b) The feasible region is the polygon bounded by the lines \(x = 1200\), \(y = 600\), \(x + y=2800\), \(x = 0\) (but restricted by \(x\geq1200\)) and \(y = 0\) (but restricted by \(y\geq600\)). To graph it, plot the lines as described above and shade the intersection region.