Lesson 2 Pet Sitters Develop Understanding

Learning Focus

Represent linear constraints.

How can I represent all possible solutions to a situation that is limited in different ways by various resources or constraints?

Open Up the Math: Launch, Explore, Discuss

Carlos and Clarita are making a list of some of the issues they need to consider as part of their business plan to care for cats and dogs while their owners are on vacation.

Space: Cat pens will require $6 {ft}^{2}$ of space, while dog runs require $24 {ft}^{2}$ . Carlos and Clarita have up to $360 {ft}^{2}$ available in the storage shed for pens and runs, while still leaving enough room to move around the cages.
Start-up costs: Carlos and Clarita plan to invest much of the $$ 1, 280$ they earned from their last business venture to purchase cat pens and dog runs. It will cost $$ 32$ for each cat pen and $$ 80$ for each dog run.

Of course, Carlos and Clarita want to make as much money as possible from their business, so they are trying to determine how many of each type of pet they should plan to accommodate. They plan to charge $$ 8$ per day for boarding each cat and $$ 20$ per day for each dog.

After surveying the community regarding the pet boarding needs, Carlos and Clarita are confident that they can keep all of their boarding spaces filled for the summer.

So, the question is, how many of each type of pet should they prepare for? Their dad has suggested the same number of each, perhaps $12$ cats and $12$ dogs. Carlos thinks they should plan for more dogs, since they can charge more. Clarita thinks they should plan for more cats since they take less space and time, and therefore, they can board more.

1.

What do you think? What recommendations would you give to Carlos and Clarita, and what argument would you use to convince them that your recommendation is reasonable?

Ready for More?

Try a different, and perhaps more challenging, representation than you are currently using by considering the following questions:

How can both constraints be considered simultaneously?

How might you represent all the possible combinations of cats and dogs that Carlos and Clarita can take care of on a graph? Symbolically?

Takeaways

In Pet Sitters, the statements about start-up costs and space are called constraints because .

Viable options for the space constraint include , because .

The Pet Sitters context can be modeled with a system of linear inequalities, because .

In the Pet Sitters context, Carlos and Clarita want to maximize their profit. The statement about profit is not a constraint because,.

Instead, profit is the result of .

Vocabulary

constraint
system of inequalities
viable, non-viable options
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we encountered a situation in which decisions were limited by certain constraints. We tested combinations of quantities to see if they fit one, both, or neither constraint. Although we didn’t resolve the situation today, we recognized the goal of this type of work: to make decisions within the parameters of the constraints that will maximize (or minimize) a quantity, such as profit.

Retrieval

1.

Graph the equation and determine if $(2, 7)$ is a solution to the equation, $y = - 2 x + 7$ .

2.

Find another coordinate that is a solution to the equation $y = - 2 x + 7$ .

3.

Graph the system of equations and find the point where they intersect. Verify that the point of intersection is a solution for both equations.

${\begin{cases} y = - 3 x + 6 \\ y = 2 x - 9 \end{cases}$

Point of intersection: