Summer Business!
Rationale
This task allows the student to see how linear programming can be used to maximize earnings in real world small business applications.
Instructional Task
When summer approaches, Desmond and Terrance look forward to baseball games, movies, and weekends at the local amusement park. Because all of these things require money for admission, Desmond and Terrance decide to start a lawn service to earn money.
Desmond can devote 10 hours a week to this new venture, while Terrance can devote 4 hours. The boys realize it takes less time to trim the edge than to mow, so they decide that Desmond can do all the mowing and Terrance can do the edging. After surveying the neighborhood, they determine that their clients will fall into one of two categories: standard-sized interior lots or large corner lots.
From working on their own yards the boys know it will take about an hour to mow and half an hour to edge a standard-sized yard. A larger yard will take 45 minutes to edge and 2 hours to mow.
Based on research he has done in his neighborhood, Desmond wants to set prices at $20 per large yard and $15 for each standard-sized yard. Each price is the total price that includes edging and mowing.
- Use the table below to organize the information from paragraphs two and three in the situation above.
x (number of
standard-sized
yards)y (number
of large
yards)Constraint Number of
hours mowingNumber of
hours edging - Using the information in the table to assist you, write a system of inequalities to represent all the constraints that define the boys’ problem situation.
- Terrance and Desmond will need an objective function to describe the goal of the investigation. In this case, the goal is to maximize the boys’ earning through their lawn service. Write an objective function that describes the amount of money, P, they will earn in one week.
- Graph the system of inequalities representing the constraints and shade the feasible region—the region representing all the solutions to the system.
- What are the vertices of your feasible region? Explain how you found these.
- Use the vertices (from question 5) and the earnings equation to find the maximum Terrance and Desmond can earn in one week with the given constraints.
- How many and which size yards will Terrance and Desmond need to mow and edge to maximize their earnings?
Discussion/Further Questions/Extensions
Linear Programming:
In general, the maximum and minimum values of an objective function will always occur at one of the corner points of the feasible region. This leads to a general procedure for solving linear programming problems:
- Graph the system of linear inequalities and find the region of feasible solutions, identified by overlapped shading.
- Determine the coordinates of the vertices of that region by finding the solutions to the related systems of equations.
- Evaluate the objective function at each vertex. The smallest and largest of those values are the minimum and maximum values of the function, respectively.
Software:
If dynamic geometry software is available, show the Linear Programming model sketch. In it, participants see a feasible region with a possible income function. Drag the income line so that it is inside the feasible region and maximizes the income. What point of the feasible region lies on the line?
Sample Solutions
- Use the table below to organize the information from paragraphs two and three in the situation above.
x (number of
standard-sized
yards)y (number
of large
yards)Constraint Number of
hours mowing1x 2y ≤ 10 Number of
hours edging
≤ 4 - Using the information in the table to assist you, write a system of inequalities to represent all the constraints that define the boys’ problem situation.
x + 2y ≤ 10,
,x ≥ 0, and
y ≥ 0
- Terrance and Desmond will need an objective function to describe the goal of the investigation. In this case, the goal is to maximize the boys’ earning through their lawn service. Write an objective function that describes the amount of money, P, they will earn in one week.
P = 15x + 20y
- Graph the system of inequalities representing the constraints and shade the feasible region—the region representing all the solutions to the system.
- What are the vertices of your feasible region? Explain how you found these.
(0, 0); (0, 5); (2, 4); (8, 0)
Explanations will vary but could include:
- I solved systems of linear equations to find the intersection points.
- I looked carefully at the graph to determine intersection points of the segments, then checked my estimates by substituting values for x and y into the function rules.
- I used technology to determine intersection points.
- Use the vertices (from question 5) and the earnings equation to find the maximum Terrance and Desmond can earn in one week with the given constraints.
Using the earnings equation from question 3, P = 15x + 20y, where P represents the amount earned in dollars:
Vertex P = 15x + 20y (0, 0) $0 (0, 5) $100 (2, 4) $110 (8, 0) $120 Their maximum earnings are $120.
- How many and which size yards will Terrance and Desmond need to mow and edge to maximize their earnings?
For Terrance and Desmond to maximize their earnings, they will need to mow and edge 8 standard-sized yards and 0 large yards. Their earnings will be $120 a week.
This activity is adapted from Agile Mind, Algebra II, Topic 16, Exploring 2. Used with permission.