Lesson 5 Pampering and Feeding Time Practice Understanding

Learning Focus

Represent constraints symbolically and graphically.

What are efficient ways to write the inequalities and sketch the solution sets representing these additional constraints on feeding time and pampering time?

Open Up the Math: Launch, Explore, Discuss

Carlos and Clarita have been worried about space and start-up costs for their pet sitting business, but they realize they also have a limit on the amount of time they have for taking care of the animals they board. To keep things fair, they have agreed on the following time constraints.

  • Feeding time: Carlos and Clarita estimate that cats will require minutes twice a day—morning and evening—to feed and clean their litter boxes, for a total of minutes per day for each cat. Dogs will require minutes twice a day to feed and walk, for a total of minutes per day for each dog. Carlos can spend up to hours each day for the morning and evening feedings, but needs the middle of the day off for baseball practice and games.

  • Pampering time: The twins plan to spend minutes each day brushing and petting each cat, and minutes each day bathing or playing with each dog. Clarita needs time off in the morning for swim team and evening for her art class, but she can spend up to hours during the middle of the day to pamper and play with the pets.

1.

Write each of these additional time constraints symbolically.

Feeding time constraint:

Pampering time constraint:

2.

Shade the solution set for each constraint on separate coordinate grids.

Feeding time:

a blank 17 by 17 grid

Pampering time:

a blank 17 by 17 grid

3.

Now find the point of intersection for the two time constraints, pampering time and feeding time.

4.

What does this point mean in the context of cats and dogs?

Ready for More?

One week Carlos twisted his ankle during baseball practice, and so it required more time for him to feed and walk the dogs. Here is the feeding time constraint for that week:

  • Feeding time: Each cat will require minutes per day to feed and clean their litter boxes. Each dog will require minutes per day to feed and walk them. Carlos can spend up to hours each day doing these activities.

  • Pampering time: Clarita will spend minutes each day brushing and petting each cat, and minutes each day bathing or playing with each dog. Clarita can spend up to hours during the middle of the day to pamper and play with the pets.

1.

Find the point of intersection for these revised time constraints.

Feeding time constraint:

Pampering time constraint:

Point of intersection:

2.

What does this point mean in the context of cats and dogs?

Takeaways

Ways to determine which half plane should be shaded after graphing the boundary line for a linear inequality include:

Using different units for measuring the quantities in a constraint leads to:

A statement about a limitation or restriction in a modeling context may be eliminated as a constraint if:

Lesson Summary

In this lesson, we drew upon everything we have learned so far about modeling contexts with linear inequalities as constraints. We learned that in a system of constraints, quantities need to be measured in the same units, and that individual constraints are equivalent and produce the same solution set even if the quantities are measured in different units.

Retrieval

Rewrite the equations so that they are in slope-intercept form.

1.

2.

Rewrite the given equations so that they are in standard form.

3.

4.

5.

Solve the system of equations by substitution.