Chapter 7.5, Problem 69E

(a)

To determine

To graph: A system of inequalities for different amount of food X and food Ywith the help of prescribed plan by dietician.

Explanation of Solution

Given:

The maximum weight to hold the weight in the truck is 7500 pounds. A warehouse requires at least 50 packages to ship of 55 pounds and at least 40 bags of stone to ship of 70 pounds.

Graph:

Consider x as the amount of 55lb bags and y as the amount of 70lb bags.

The maximum weight to hold the weight in the truck is 7500 pounds. A warehouse requires at least 50 packages to ship of 55 pounds and at least 40 bags of stone to ship of 70 pounds.

As per the given information, the inequalities are set up as follows:

  $\begin{array}{l}x\ge 50\\ y\ge 40\\ 55x+70y\le 7500\end{array}$

Draw the graph with the help of lines $x=50$, $y=40$ and $55x+70y=7500$ .

Find the coordinates of the line $55x+70y=7500$.

Substitute $10$ for $x$ in the equation $55x+70y=7500$ to obtain the $y$ coordinate.

  $\begin{array}{c}55\left(10\right)+70y=7500\\ 550+70y=7500\\ 70y=7500-550\\ 70y=6950\end{array}$

Further simplify the above calculation.

  $\begin{array}{l}y=\frac{6950}{70}\\ y=\frac{695}{7}\end{array}$

Substitute $20$ for $x$ in the equation $55x+70y=7500$ to obtain the $y$ coordinate.

  $\begin{array}{c}55\left(20\right)+70y=7500\\ 1100+70y=7500\\ 70y=6400\\ y=\frac{640}{7}\end{array}$

The coordinates are $\left(10,\frac{695}{7}\right)\text{ and }\left(20,\frac{640}{7}\right)$.

Shade the region above the lines for the sign of at least and shade the region below the line for the sign of at most in three inequalities.

Now draw the graph with the help of obtained coordinates of the lines.

  

Figure 1

Interpretation:

It can be observed from the graph that the shaded region lies in the first quadrant.

(b)

To determine

To find: Two solutions from and interpret the result.

Explanation of Solution

Solutions of the system of inequalities are those points that comes under the feasible region of the graph.

The feasible region of the graph is the shaded region of the graph.

Take any two points from the shaded region of the graph.

It can be seen from the graph that the points $\left(60,50\right)$ and $\left(58,52\right)$ comes under the shaded region of the graph.

The point $\left(60,50\right)$ represents 60 is the amount of 55lb bags and 50 is the amount of 70lb bags.

The point $\left(58,52\right)$ represents 60 is the amount of 55lb bags and 50 is the amount of 70lb bags.