Chapter 2.7, Problem 13E

(a)

To determine

To find: The inequality to represent the number of day-shift workers, the night-shift workers, and the total number of workers needed.

Answer to Problem 13E

The inequality is $0.84x+0.56y\ge 1.54$ .

Explanation of Solution

Given:

Angela’s pizza is open from noon to midnight day.

Employees work for 8-hour shifts from noon to 8 P.M. or 4 P.M. to midnight.

The inequality that represent the number of day-shift workers, the night-shift workers, and the total number of workers needed as below.

Let $x$ be purely day shift workers,

And $y$ be purely night shift workers.

  $\begin{array}{l}x\ge 5\left[\text{noon to}4\right]\\ y\ge 6\left[8\text{to midnight}\right]\\ x+y\ge 14\left[4\text{to}8\right]\end{array}$

Therefore the inequality is given above.

(b)

To determine

To find: The graph the system of inequality.

Answer to Problem 13E

The graph is given below.

Explanation of Solution

The graph of the above inequality is given below:

  

In order to check the shaded region contains origin or not, check after substituting $\left(x,y\right)=\left(0,0\right)$ in the equation of inequalities satisfies or not.

  $\begin{array}{l}0\ge 5\left[\text{false}\right]\\ 0\ge 6\left[\text{false}\right]\\ 0+0\ge 14\left[\text{false}\right]\end{array}$

The shaded region does not contains origin.

Therefore, the graph is given above.

(c)

To determine

To find: The function to represent the daily cost of payroll.

Answer to Problem 13E

The function to represent the daily cost of payroll is $c\left(x,y\right)=5.50x+7.50y$ .

Explanation of Solution

The cost of payment that are paid to employees those work between noon and 4P.M is $5.50 per hour.

The cost of payment that are paid to employees those work between 4P.M and mid-night is $7.50 per hour.

So the function to represent the daily cost of payroll is given as:

  $c\left(x,y\right)=5.50x+7.50y$

Here $x$ is the number of workers worked between noon and 4P.M.

And $y$ is the number of workers worked between 4P.M and mid-night.

Therefore the function to represent the daily cost of payroll is $c\left(x,y\right)=5.50x+7.50y$ .

(d)

To determine

To find: The number of day-shift workers and night-shift workers that should be scheduled to minimise to cost.

Answer to Problem 13E

The number of cups of each type of puppy: 1.5 cups of Good Start and $0.5$ cups of Sirius.

Explanation of Solution

In order to minimise the cost, test each point in the minimizing function, that are shown on the boundary of the graph.

  $\begin{array}{c}c\left(5,9\right)=5.5\left(5\right)+7.5\left(9\right)\\ =95\\ c\left(8,6\right)=5.5\left(8\right)+7.5\left(6\right)\\ =89\end{array}$

The minimum cost attains at the point $\left(8,6\right)$ .

Therefore the number of day-shift workers is 8 and night-shift workers is 6.

(e)

To determine

To find: The minimum cost.

Answer to Problem 13E

The minimum cost is $89$ .

Explanation of Solution

The minimum cost at $\left(8,6\right)$ is calculated as:

  $\begin{array}{c}c\left(8,6\right)=5.5\left(8\right)+7.5\left(6\right)\\ =89\end{array}$

Therefore, the minimum cost is $89$ .