Chapter 7, Problem 86RE

To determine

To calculate: A dietician prescribed a special dietary plan using two different foods. Each ounce of food X contains 200 milligrams of calcium, 3 milligrams of iron, and 100 milligram of magnesium. Each ounce of food Y contains 150 milligrams of calcium, 2 milligrams of iron, and 80 milligram of magnesium. The minimum daily requirements of the diet are 800 milligrams of calcium, 10 milligrams of iron, and 200 milligram of magnesium. Write and graph a system of inequalities that describes the different amount of food X and food Y that can be prescribed.

Answer to Problem 86RE

The system of inequality is:

  $\begin{array}{l}200x+100y\ge 800\\ 3x+2y\ge 10\\ 100x+80y\ge 200\\ x\ge 0,y\ge 0\end{array}$

Explanation of Solution

Let x be amount of food X and y be amount of food Y

The minimum calcium required is 800 milligram, food X contain 200 milligram and food Y contain 100 milligram, so

  $200x+100y\ge 800$

The minimum iron required is 10 milligram, food X contain 3 milligram and food Y contain 2 milligram, so

  $3x+2y\ge 10$

The minimum magnesium required is 200 milligram, food X contain 100 milligram and food Y contain 80 milligram, so

  $100x+80y\ge 200$

Both foods are always either equal or greater than 0,

So $x\ge 0,y\ge 0$

For the graph of the equation $200x+100y=800$

Make a table

  $\begin{array}{cccccc}x& 0& 1& 2& 3& 4\\ y& 8& 6& 4& 2& 0\end{array}$

Plot the points on the graph and join the lines

  

For the graph of the equation $3x+2y=10$

Make a table

  $\begin{array}{cccccc}x& 0& 2& 4& 6& -2\\ y& 5& 2& -1& -4& 8\end{array}$

Plot the points on the graph and join the lines

  

For the graph of the equation $100x+80y=200$

Make a table

  $\begin{array}{cccccc}x& 0& 1& 2& 3& -1\\ y& 2.5& 1.25& 0& -1.25& 3.75\end{array}$

Plot the points on the graph

Plot all the equations in one graph keeping in mind that $x\ge 0,y\ge 0$

With the area as $200x+100y\ge 800,3x+2y\ge 10\text{and}100x+80y\ge 200$

  

Conclusion:

  $\begin{array}{l}200x+100y\ge 800\\ 3x+2y\ge 10\\ 100x+80y\ge 200\\ x\ge 0,y\ge 0\end{array}$