Chapter 7.5, Problem 67E

(a)

To determine

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

Explanation of Solution

Given:

According to the dietician, food X contains 180mg of calcium, 6mg of iron and 220 mg of magnesium in each ounce and food Y contains 100mg of calcium, 1mg of iron and 40mg of magnesium in each ounce. The diet requires at least 1000mg of calcium, 18mg of iron and 400mg of magnesium.

Graph:

Consider x as the amount of food $X$ and y as the amount of food $Y$ .

Each ounce of food $X$ contains 180mg of calcium and each ounce of food $Y$ contains

100mg of calciumbut the diet requires at least 1000mg of calcium.

Therefore, set up the inequality as follows:

  $\begin{array}{c}180x+100y\ge 1000\\ 9x+5y\ge 500\end{array}$

Each ounce of food $X$ contains 6mg of iron and each ounce of food $Y$ contains

1mg of iron but the diet requires at least 18mg of iron.

Therefore, set up the inequality as follows:

  $6x+y\ge 18$

Each ounce of food $X$ contains 220 mg of magnesium and each ounce of food $Y$ contains

40mg of magnesium but the diet requires at least 400mg of magnesium.

Therefore, set up the inequality as follows:

  $\begin{array}{c}220x+40y\ge 400\\ 11x+2y\ge 20\end{array}$

Draw the graph with the help of lines $9x+5y=500$, $6x+y=18$ and $11x+2y=20$ .

Find the coordinates of the line $9x+5y=500$.

Substitute $10$ for $x$ in the equation $9x+5y=500$ to obtain the $y$ coordinate.

  $\begin{array}{c}9\left(10\right)+5y=500\\ 90+5y=500\\ 5y=500-90\\ 5y=410\\ y=\frac{410}{5}\end{array}$

Substitute $20$ for $x$ in the equation $9x+5y=500$ to obtain the $y$ coordinate.

  $\begin{array}{c}9\left(20\right)+5y=500\\ 180+5y=500\\ 5y=500-180\\ 5y=380\\ y=\frac{380}{5}\\ y=76\end{array}$

The coordinates are $\left(10,\frac{410}{5}\right)\text{ and }\left(20,76\right)$.

Substitute $1$ for $x$ in the equation $11x+2y=20$ to obtain the $y$ coordinate.

  $\begin{array}{c}11x+2y=20\\ 11\left(1\right)+2y=20\\ 11+2y=20\\ 2y=9\\ y=4.5\end{array}$

Substitute $5$ for $x$ in the equation $11x+2y=20$ to obtain the $y$ coordinate.

  $\begin{array}{c}11\left(5\right)+2y=20\\ 55+2y=20\\ 2y=-35\\ y=-\frac{35}{2}\end{array}$

The coordinates are $\left(1,4.5\right)\text{ and }\left(5,-\frac{35}{2}\right)$.

Substitute $1$ for $x$ in the equation $6x+y=18$ to obtain the $y$ coordinate.

  $\begin{array}{c}6\left(1\right)+y=18\\ y=18-6\\ y=12\end{array}$

Substitute $5$ for $x$ in the equation $6x+y=18$ to obtain the $y$ coordinate.

  $\begin{array}{c}6\left(5\right)+y=18\\ 30+y=18\\ y=-12\end{array}$

The coordinates are $\left(1,12\right)\text{ and }\left(5,-12\right)$.

Shade the region above the lines since the sign is of at least 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 line $6x+y=18\text{ and }5x+5y=500$ intersect at point $\left(19.5,135\right)$ .

(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 $\left(10,300\right)$ and $\left(20,300\right)$ comes under the shaded region of the graph.

The point $\left(10,300\right)$ represents 10 ounces of Food $X$ and 300 ounces of food $Y$ .

The point $\left(20,300\right)$ represents 20 ounces of Food $X$ and 300 ounces of food $Y$ .