Chapter 3.3, Problem 27PPS

(a)

To determine

To find: the number of each type of package that should be placed on a train car to maximize revenue.

Answer to Problem 27PPS

The maximum package is $800$ . Therefore, there are $160$ small packages and $0$ large packages.

Explanation of Solution

Given:

Car on a freight can hold $4200$ pounds of cargo and has capacity of $480$ cubic feet.

Freight service handle two types of packages which weighs $25$ pounds and are $3$ cubic feet each,

And a large which are $50$ pounds and are $5$ cubic feet each. The freight service charge $\$5$ for each small package and $\$8$ for each large package.

Concept used:

Graphing the inequalities, it will graph the ordinary linear functions just like it is done before. The difference is that a solution to the inequality is not the drawn line but coordinate plane that satisfy the inequality.

The boundary line is dashed for $<\text{or >}$ and solid line for $\le \text{or }\ge$ .

Calculation:

Suppose $x$ be the number of small packages and $y$ be the number of larger packages.

Then, the optimizing is written as follows.

  $f\left(x,y\right)=5x+8y.$

The constraints are:

  $\begin{array}{l}25x+50y\le 4200.\\ 3x+5y\le 480.\\ x\ge 0.\\ y\ge 0.\end{array}$

The graph is shown below.

The vertices of the feasible region are $\left(0,0\right),\left(0,84\right),\left(160,0\right)\text{ and }\left(120,24\right)$ .

The table is created as shown below:

$\left(x,y\right)$	$f\left(x,y\right)=5x+8y$
$\left(120,24\right)$	$5\left(120\right)+8\left(24\right)=792$
$\left(160,0\right)$	$5\left(160\right)+8\left(0\right)=800$
$\left(0,84\right)$	$5\left(0\right)+8\left(84\right)=336$
$\left(0,0\right)$	$5\left(0\right)+8\left(0\right)=0$

The maximum package is $800$ . Therefore, there are $160$ small packages and $0$ large packages.

(b)

To determine

To find: the maximum revenue per train car.

Answer to Problem 27PPS

It is $\$800$ .

Explanation of Solution

Given:

The Car on a freight can hold $4200$ pounds of cargo and has capacity of $480$ cubic feet.

Freight service handle two types of packages which weighs $25$ pounds and are $3$ cubic feet each,

And a large which are $50$ pounds and are $5$ cubic feet each. The freight service charge $\$5$ for each small package and $\$8$ for each large package.

Concept used:

Graphing the inequalities, it will graph the ordinary linear functions just like it is done before. The difference is that a solution to the inequality is not the drawn line but coordinate plane that satisfy the inequality.

The boundary line is dashed for $<\text{or >}$ and solid line for $\le \text{or }\ge$ .

Calculation:

The maximum package is $800$ .

The maximum revenue per train car is $800$ dollars.

Hence, it is $\$800$ .

(c)

To determine

To find: whether the maximizing revenue is necessary a best thing company does and explain.

Answer to Problem 27PPS

It is not necessary to maximize the revenue.

Explanation of Solution

Given:

The Car on a freight can hold $4200$ pounds of cargo and has capacity of $480$ cubic feet.

Freight service handle two types of packages which weighs $25$ pounds and are $3$ cubic feet each,

And a large which are $50$ pounds and are $5$ cubic feet each. The freight service charge $\$5$ for each small package and $\$8$ for each large package.

Concept used:

Graphing the inequalities, it will graph the ordinary linear functions just like it is done before. The difference is that a solution to the inequality is not the drawn line but coordinate plane that satisfy the inequality.

The boundary line is dashed for $<\text{or >}$ and solid line for $\le \text{or }\ge$ .

Calculation:

the maximizing the revenue is not necessary because of the revenue is maximized the company will not deliver any large packages and customers will large packages to ship will probably choose another carrier.

Hence, it is not necessary to maximize the revenue.