Linar Programing Questions

Question

linear programming problem 1

Sandy Wich runs a mobile lunch cart/land where he still sells hot dogs and hamburgers at the TCC Picnic on Saturday. Sandy has at most 100 hots dogs to sell and at most 75 hamburgers to sell. Each Hot dog and $2.00 profit on each hamburger.

How many hot dogs and how many hamburgers should sandy bring to maximize his profit and What is that profit amount if he bring that many of the sandwiches sell them all.

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.

Answer 1

Step 1: Identify Variables

Let x be the number of hot dogs sold by Sandy Wich and y be the number of hamburgers sold by him.

Step 2: Create the Objective Function

The objective of Sandy Wich is to maximize his profit. Each hot dog yields $1 profit, and each hamburger yields $2 profit. Therefore, the objective function is:

Profit (P) = 1x + 2y

Step 3: Write the Constraints

Sandy Wich has at most 100 hot dogs and at most 75 hamburgers to sell. These are the constraints on the number of sandwiches sold. Therefore, the system of inequalities is:

x ≤ 100 (maximum number of hot dogs)

y ≤ 75 (maximum number of hamburgers)

Both x and y should be non-negative:

x ≥ 0

y ≥ 0

Step 4: Graph the System of Inequalities

We can graph the system of inequalities by plotting the boundary lines for each inequality and shading the feasible region. The feasible region will be the polygon formed by the intersection of all the inequalities.

To graph the system of inequalities, we can plot the line x = 100 (vertical line passing through (100,0)), the line y = 75 (horizontal line passing through (0,75)), and the x and y axes. Then, we shade the region that satisfies all of the inequalities (the feasible region).

Step 5: Identify the Corner or Vertex Point of the Polygonal Set

The corner points or vertex points of the polygonal set are the points where the boundary lines intersect. In this case, the polygonal set has four corner points: (0,0), (0,75), (100,0), and (50,25).

Step 6: Use the Vertex Points to Find the Maximum Value of the Objective Function

We can use the objective function to evaluate the profit at each of the vertex points to find the maximum value. The profit at each vertex point is:

(0,0) -> Profit (P) = 1(0) + 2(0) = $0

(0,75) -> Profit (P) = 1(0) + 2(75) = $150

(100,0) -> Profit (P) = 1(100) + 2(0) = $100

(50,25) -> Profit (P) = 1(50) + 2(25) = $100

Therefore, the maximum profit Sandy Wich can earn is $150 if he sells only hamburgers and no hot dogs. He should bring 0 hot dogs and 75 hamburgers to the picnic to maximize his profit.

Answer: Sandy Wich should bring 0 hot dogs and 75 hamburgers to the TCC Picnic on Saturday to maximize his profit. His maximum profit will be $150 if he sells only hamburgers and no hot dogs.