Chapter 3.3, Problem 33E

### Finite Mathematics for the Manager...

12th Edition
Soo T. Tan
ISBN: 9781337405782

Chapter
Section

### Finite Mathematics for the Manager...

12th Edition
Soo T. Tan
ISBN: 9781337405782
Textbook Problem

# The problems in Exercises 31-51 correspond to those in Exercises 1-21, Section 3.2. Use the results of your previous work to help you solve these problems. 31. PRODUCTION SCHEDULING Kane Manufacturing has a division that produces two models of fireplace grates, model A and model B. To produce each model A grate requires 3 lb of cast iron and 6 min of labor. To produce each model B grate requires 4 lb of cast iron and 3 min of labor. The profit for each model A grate is $2.00 , and the profit for each model B grate is$ 1.50 If 1000 lb of cast iron and 20 labor-hours are available for the production of fireplace grates per day, how may grates of each model should the division produce to maximize Kane's profit? What is the optimal profit?

To determine

The number of grates of each model that should be produced in per day to maximize the Kane`s profit and optimal profit.

Explanation

Given:

Tabulate the given information.

 Models Cast iron Time (in min) Profit ( in $) A 3â€‰lb 6m$2.00 B 4â€‰lb 3m \$1.50

Approach:

Assume x is the no. of units to manufacture the model A and y is the no. of units to manufacture the model B. So, the total profit P (in dollars) is P=2x+1.5y and this is the objective function to be maximized.

The total amount for Cast iron is given by 3x+4y and must not exceed 1000â€‰lb. So, the inequality is,

3x+4yâ‰¤1000

The total amount of time is given by 6x+3y and must not exceed 1200.So, the inequality is,

6x+3yâ‰¤1200

As neither x nor y will be negative so,

xâ‰¥0yâ‰¥0

Calculation:

Consider the given inequalities.

3x+4yâ‰¤10006x+3yâ‰¤1200

Replace the inequality sign with equality sign in the above linear inequalities and write them in intercept form.

3x+4y=10003x1000+y250=1

So, the coordinates are (33.33,0) and (0,250).

Also,

6x+3y=1200x200+y400=1

So, the coordinates are (200,0) and (0,400)

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