Chapter 1.8, Problem 23E

a.

To determine

To find: linear inequality for x and y that could be baked in 8 hour shift.

Answer to Problem 23E

  $480\ge 8x+10y$

Explanation of Solution

Given information:

Number of batches of chocolate cookies = x

Number of batches of vanilla cookies = y

Baking timing for x = 8 minutes

Baking timing for y = 10 minutes

Calculation:

Convert 8 hours into minutes as follows:

  $\begin{array}{c}8\text{ hours}=8\text{ hours}\left(\frac{60\text{minutes}}{1\text{ hour}}\right)\\ =480\text{ minutes}\end{array}$

The time taken to bake x and y is 480 minutes.

Therefore, the inequality will be $480\ge 8x+10y$ .

b.

To determine

To graph: the inequality.

Explanation of Solution

Given information:

  $480\ge 8x+10y$

Calculation:

Rearrange the inequality in terms of y as follows:

  $\begin{array}{c}480\ge 8x+10y\\ 480-8x\ge 10y\\ \frac{480}{10}-\frac{8}{10}x\ge y\\ 48-0.8x\ge y\end{array}$

Create a table by plugging the different values x in the line equation $y=48-0.8x$ to evaluate the value for y.

$x$	$y$
$-$ 40	80
$-$ 20	64
0	48
20	32
40	16
60	0
80	$-$ 16

Since the inequality has greater than or equal to sign $(\ge )$ , solid line will be used to graph a line $y=48-0.8x$ .

Put the test points $\left(60,1\right)$ and $\left(0,0\right)$ in the inequality and check the true test point as follows:

Test point   $\left(x,y\right)$	Inequality,   $48-0.8x\ge y$	True/False test point
$\left(0,0\right)$	$\begin{array}{l}48-0.8\left(0\right)\ge 0\\ 48\ge 0\end{array}$	True
$\left(60,1\right)$	$\begin{array}{c}48-0.8\left(60\right)\ge 1\\ 48-48\ge 1\\ 0\ge 1\end{array}$	False

The graph for the given inequality is shown below.

c.

To determine

To find: the three combinations of x and y for which the inequality is satisfied.

Answer to Problem 23E

  $\begin{array}{l}\left(0,48\right)\\ \left(10,40\right)\\ \left(60,0\right)\end{array}$

Explanation of Solution

Given information:

Number of batches of chocolate cookies = x

Number of batches of vanilla cookies = y

Baking timing for x = 8 minutes

Baking timing for y = 10 minutes

The inequality found in (a) is $480\ge 8x+10y$

Calculation:

Make a table by substituting three different values of x and evaluate the value of y from $480=8x+10y$ as follows:

x	y
0	48
10	40
60	0

Therefore, the three combinations satisfying the inequality are $\left(0,48\right),\left(10,40\right)$ and $\left(60,0\right)$ .

d.

To determine

To find: how the given problems are solved.

Answer to Problem 23E

By using complex computer programs and system of inequalities.

Explanation of Solution

Given information:

Manufacture’s production problem is how much each product should be assigned to each machine. It involves as many as 150 products, 218 facilities, 10 plants and 127 customer zones.

Calculation:

Manufacture’s production problem can be solved by using complex computer programs and system of inequalities.