problem for linear Program

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 the x and y variables.

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 minimum cost

and give your answer completely in sentences.

Please explain step by step and graphing as well.

Answer 1

You use the information in the situation to arrive at the Objective Function which is the profit for any given day. You also use the information to arrive at the constraints for the number of scientific and graphing calculators.

x: # of Scientific Calculators

y: # of Graphing Calculators

Objective Function: Profit = -5x + 9y

Constraints

x > 70 x < 100

y > 50 y < 80

x + y > 130

You graph each inequality on the coordinate plan shading each one correctly and then the overlapping shaded regions are the feasibility solution set.

To find the intersection points, solve each system of the two lines individually for example

x = 70 and y = 80 gives the intersections point of (70, 80)

x + y = 130 and x = 70 gives the intersection (70, 60)

You asked for the minimum cost; however, there is no information about cost in the problem; therefore, I cannot provide an answer about the minimum cost.

To find the maximum profit, you evaluate the Objective Function at each intersection point.

@ (70, 60) Profit = -5(70) + 9(60) = 190

@ (80, 50) Profit = -5(80) + 9(50) = 50

@ (100, 50) Profit = -5(100) + 9(50) = -50

@ (100, 80) Profit = -5(100) + 9(80) = 220

@ (70, 80) Profit = -5(70) + 9(80) = 370

Therefore, the maximum profit will happen when 70 scientific calculators and 80 graphing calculators are shipped and sold.