Linear Program Word Questions

Question

Problem 4

A calculator company produces a scientific Calculator and a graphing calculator. Long-term projections indicate an expected demand in a market of at least 70 scientific and 50 graphing calculators each day. Because of limitations on production capacity for this market no more than 100 scientific and 80 graphing calculators can be made daily. To satisfy a shipping contract, a total of at least 130 calculators must be shipped each day to this market. Because of the cost of production material and labor and the price the calculators must be sold, the following profit is as follows for the day: For every scientific calculator shipped and sold there is a -$5 profit and for every graphing calculator shipped and sold there is a +$9 profit. Assume all calculators shipped are sold.

To satisfy their contract and given the inequality parameters, how many scientific calculators and how many scientific calculators should be made and sent, and what is the maximum profit for the day that can be achieved?



Including

Identify your variable.

create your functions formula with these variables.

Write your system of inequalities that make up your constraints.

Graph your system of inequalities.

Show your polygonal feasibility solutions set.

Identify the corner or vertex point of your polygonal set.

use these points in your functions formula to find the maximum value.

give your answer completely in sentences.

Please explain step by step and graphing as well.

Answer 1

Variable:

Let x be the number of scientific calculators produced and y be the number of graphing calculators produced.

Objective Function:

The objective is to maximize profit. Profit for scientific calculator is -$5, and for a graphing calculator, it is +$9. Therefore, the objective function is:

z = -5x + 9y

Constraints:

Long-term projections indicate an expected demand in a market of at least 70 scientific and 50 graphing calculators each day:

x ≥ 70

y ≥ 50

Because of limitations on production capacity for this market, no more than 100 scientific and 80 graphing calculators can be made daily:

x ≤ 100

y ≤ 80

To satisfy a shipping contract, a total of at least 130 calculators must be shipped each day to this market:

x + y ≥ 130

Graphing:

To graph the constraints, we can plot the lines: x = 70, y = 50, x = 100, y = 80, and x + y = 130.

Feasible Region:

To find the feasible region, we need to shade the region that satisfies all the constraints. It is the shaded region in the above graph.

Vertex Points:

The vertex points of the feasible region are (70, 60), (70, 80), (100, 30), and (100, 80).

Finding the Maximum Profit:

We can substitute each vertex point into the objective function and find the maximum value:

At (70, 60): z = -5(70) + 9(60) = 210

At (70, 80): z = -5(70) + 9(80) = 330

At (100, 30): z = -5(100) + 9(30) = -170

At (100, 80): z = -5(100) + 9(80) = 320

Therefore, the maximum profit of $330 can be achieved by producing 70 scientific calculators and 80 graphing calculators.