Chapter 11, Problem 61RE

To determine

To find: The number of engines produced per week.

Answer to Problem 61RE

The number of engines produced per week are $35$ gasoline engines and $15$ diesel engines must be produced to minimize the cost.

Explanation of Solution

Given information:

The given, cost of gasoline engine is $\$450$ , and diesel engine cost is $\$550$ .

Calculation:

Calculate the number of engines,

  $x$ represents the number of gasoline engines and $y$ represents the number of diesel engines.

The factory is obligated to produce at least $20$ gasoline engines, and the $15$ diesel engines per week.

The first two constraints are $x\ge 20\text{ and }y\ge 15$ .

The factory cannot produce more than $60$ gasoline engines, and $40$ diesel engines.

The next two constant are $x\le 60\text{ and }y\ge 15$ .

The total number of engines to be produce must be $x+y\ge 50$ .

If the $C$ donate the total cost,

Therefore, $C=450x+550y$ , which is objective function.

Minimize the objective function, the graph $\left(1\right)$ given inequalities and label the corner point..

  

Graph $\left(1\right)$

List the corner point and the corresponding value of the objective function in the table.

  

The minimize value of the objective function is $24,000$ , and occurs at the point $\left(35,15\right)$ .

Therefore, $35$ gasoline engines and $15$ diesel engines must be produced to minimize the cost.