Chapter 7, Problem 84RE

a.

To determine

Write a system of inequalities for dietary supplements.

Answer to Problem 84RE

  $\boxed{\begin{array}{l}12x+15y\ge 300\\ 10x+20y\ge 280\\ 20x+15y\ge 300\end{array}}$

Explanation of Solution

Given information:

A dietician designs a special dietary supplement using two different foods. Each ounce of food $X$ contains $12$ units of calcium, $10$ units of iron, and $20$ units of vitamin $B$ . Each ounce of food $Y$ contains $15$ units of calcium, $20$ units of iron, and $12$ units of vitamin $B$ . The minimum daily requirements of the diet are $300$ units of calcium, $280$ units of iron, and $300$ units of vitamin $B$ .

Write a system of inequalities describing the different amounts of food $X$ and food $Y$ that can be used.

Calculation:

Let take $x$ and $y$ be the amount of food $X$ and $Y$ .

Each ounce of food $X$ contains $12$ units of calcium and food $Y$ contains $15$ units of calcium and

The minimum daily requirements of the diet are $300$ units of calcium

So,

  $12x+15y\ge 300$

Each ounce of food $X$ contains $10$ units of iron and food $Y$ contains $20$ units of iron The minimum daily requirements of the diet are $280$ units of iron.

  $10x+20y\ge 280$

Each ounce of food $X$ contains $20$ units of vitamin $B$ and food $Y$ contains $12$ units of vitamin $B$

  $20$ units of iron The minimum daily requirements of the diet are $300$ units of iron.

  $20x+15y\ge 300$

Hence, the system of inequalities is,

  $\boxed{\begin{array}{l}12x+15y\ge 300\\ 10x+20y\ge 280\\ 20x+15y\ge 300\end{array}}$

b.

To determine

Write a system of inequalities for dietary supplements.

Answer to Problem 84RE

The graph has shown the results.

Explanation of Solution

Given information:

A dietician designs a special dietary supplement using two different foods. Each ounce of food $X$ contains $12$ units of calcium, $10$ units of iron, and $20$ units of vitamin $B$ . Each ounce of food $Y$ contains $15$ units of calcium, $20$ units of iron, and $12$ units of vitamin $B$ . The minimum daily requirements of the diet are $300$ units of calcium, $280$ units of iron, and $300$ units of vitamin $B$ .

Sketch a graph of the region in part (a).

Calculation:

Find the point of intersection on the lines,

  $\begin{array}{l}12x+15y=\mathrm{300....}\left(1\right)\\ 20x+12y=\mathrm{300.....}\left(2\right)\end{array}$

Multiply equation $\left(1\right)$ by $5$ and $\left(2\right)$ by $3$ and subtract them.

  $\begin{array}{l}\cancel{6}0x+75y-\bcancel{6}0x-36y=1500-900\\ 39y=600\\ y=\frac{200}{13}\end{array}$

Put $y=\frac{200}{13}$ in equation $\left(1\right)$ .

  $\begin{array}{l}12x+15\times \frac{200}{13}=300\\ x=\frac{75}{13}\end{array}$

Find the point of intersection on the lines,

  $\begin{array}{l}12x+15y=\mathrm{300....}\left(1\right)\\ 10x+20y=\mathrm{280....}\left(2\right)\end{array}$

Multiply equation $\left(1\right)$ by $10$ and $\left(2\right)$ by $12$ and subtract them.

  $\begin{array}{l}120x+150y-120x-240y=3000-3360\\ -90y=-360\\ y=4\end{array}$

Put $y=4$ in equation $\left(1\right)$ .

  $\begin{array}{l}12x+15\times 4=300\\ 12x=240\\ x=20\end{array}$

The graph of the region is,

  

Hence, the graph has shown the results.

c.

To determine

Write a system of inequalities for dietary supplements.

Answer to Problem 84RE

The points are $\left(10,20\right)$ and $\left(20,10\right)$ .

Explanation of Solution

Given information:

A dietician designs a special dietary supplement using two different foods. Each ounce of food $X$ contains $12$ units of calcium, $10$ units of iron, and $20$ units of vitamin $B$ . Each ounce of food $Y$ contains $15$ units of calcium, $20$ units of iron, and $12$ units of vitamin $B$ . The minimum daily requirements of the diet are $300$ units of calcium, $280$ units of iron, and $300$ units of vitamin $B$ .

Find two solutions to the system and interpret their meanings in the context of the problem.

Calculation:

The point $\left(10,20\right)$ and $\left(20,10\right)$ are two solutions of the system as they satisfy the inequalities.

The point $\left(10,20\right)$ means The minimum daily requirements are obtained by mixing $10$ ounces of food $X$ and $20$ ounces of food $Y$ .

Hence, the points are $\left(10,20\right)$ and $\left(20,10\right)$ .