Q. 61 - Precalculus Enhanced with Graphing Utilities

Question

A factory produces gasoline engines and diesel engines. Each week the factory is obligated to deliver at least $20$ gasoline engines and at least $15$ diesel engines. Due to physical limitations, however, the factory cannot make more than $60$ gasoline engines nor more than $40$  diesel engines in any given week. Finally, to prevent layoffs, a total of at least $50$ engines must be produced. If gasoline engines cost $\backslash (450$ each to produce and diesel engines cost $\backslash )550$ each to produce, how many of each should be produced per week to minimize the cost? What is the excess capacity of the factory; that is, how many of each kind of engine is being produced in excess of the number that the factory is obligated to deliver?

Step-by-Step Solution

Answer

To minimize the cost $35$ gasoline engines and $15$ diesel engines must be produced. The excess capacity of the factory is $15$ gasoline engines.

Step 1. Given

Each factory should deliver at least $20$ gasoline and $15$ diesel engines.

The factory should not produce more than $60$ gasoline engines and $40$ diesel engines.

Totally at least $50$ engines must be produced.

The cost of gasoline engine is $\$450$ and the cost of diesel engine is $\$550$

Step 2. Assume the variables.

Let x denote the number of gasoline engines and y denote the number of diesel engines.

The factory is required to produce at least $20$ gasoline and $15$ diesel engines.

So, $x\ge 20; y\ge 15$

And also the factory should not produce more than $60$ diesel engines and $40$ gasoline engines.

So, $x\le 60; y\le 40$

Step 3. Find the objective function.

the total number of engines produced must be greater than or equal to $50$,

$x+y\ge 50$

The cost of each gasoline is $\$450$ and that of diesel is $\$550$, so the objective function is,

$C\left(x\right)=450x+550y$

Step 4. Graph the inequalities

The graph of the constraints (the feasible points) is illustrated:

Step 5. Tabulate the corner points

We list the corner points and evaluate the objective function at each.

Corner points (x,y)	Value of the objective function $C\left(x\right)=450x+550y$
$(35,15)$	$$\begin{gathered} C\left(x\right)=450\left(35\right)+550\left(15\right) \\ =24,000 \end{gathered}$$
$(60,15)$	$$\begin{gathered} C\left(x\right)=450\left(60\right)+550\left(15\right) \\ =35,250 \end{gathered}$$
$(60,40)$	$$\begin{gathered} C\left(x\right)=450\left(60\right)+550\left(40\right) \\ =49,000 \end{gathered}$$
$(20,40)$	$$\begin{gathered} C\left(x\right)=450\left(20\right)+550\left(40\right) \\ =31,000 \end{gathered}$$
$(20,30)$	$$\begin{gathered} C\left(x\right)=450\left(20\right)+550\left(30\right) \\ =25,500 \end{gathered}$$

Step 6. Minimize the cost.

From the table, we can observe that the minimum cost is $\$24,000$ and occurs at $x=35; y=15$

Step 7. Excess capacity of the factory

Each factory should deliver at least $20$ gasoline and $15$ diesel engines.

Excess gasoline engine is $=35-20$

                                            $=15$

While the diesel must be produced as $15$ engines and it is produced.