Lesson 9: Constraints

Today’s Standard MCC9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. (Limit to linear equations.)

Important Vocabulary Constraint: A limitation or restriction Example: Length can’t be negative, we can’t have ½ or .5 of a person Discrete Values: Can only take certain values (data you count) Example: The number of students in a class (you can't have half a student). Continuous Values: Can take any value (data you measure) Example: A person's height: could be any value (within the range of human heights), not just certain fixed heights, time in a race: you could even measure it to fractions of a second.

Example 1: Indentifying and Representing Constraints on Variables The current population of Morrow High School 1,750. The enrollment increases at a rate of 55 students each year. Write an equation to represent enrollment at Morrow High School in terms of years. 1. What are our variables? 2. Do these variables have any constraints? If so, what are they and why?

Guided Practice Initially a pool contains 350 gallons of water. A hose is placed in the pool and the water is turned on. The hose adds 5.2 gallons of water per minute. Model the total amount, V, of water in the pool for x, the number of minutes the hose has been on. Explain your answer in the box to the right. The cost of a layer cake is $12.96 plus $3.55 per layer. If Paul has only $30 to spend on a cake, what is the most number of layers he can afford? A yearbook company promises to give their freshman class a picnic if they spend at least $6,000 on yearbooks and class rings. Each yearbook costs $70 and each class ring costs $150. Write an inequality to represent how many yearbooks and class rings the freshman class must buy in order to get their picnic.

Example 2: Estimating and Approximating Based on Constraints You are looking to rent a car to go to Macon and back. Come and Go Car Rental charges a base rate of $25 dollars and $0.50 a mile. How many miles can you drive if you have $113? 1. Write an equation to help you answer your question. 2. What constraints are placed on the variables? 3. Solve to answer the question. Should you round up or down and how do you know?

Guided Practice The current population of Morrow High School 1,750. The enrollment increases at a rate of 55 students each year. Write an equation to represent enrollment at Morrow High School in terms of years. How many years have passed if the enrollment is 3,500? Linwood Bowser is earning money over the summer. He starts with $30, and saves $40 each week. How many weeks does he need to save in order to have $600? Keep in mind that Linwood’s summer is only 16 weeks long. The cost of a layer cake is $12.96 plus $3.55 per layer. If Paul has only $30 to spend on a cake, what is the most number of layers he can afford? Jackie is trying to make at least $1,000 this summer. How many weeks does Jackie have to work if she already has $100 saved and will be making about $70 each week?

Example 3: Eliminating Answer Choices Based on Constraints. The area of a rectangle is 42 square feet. The length is 1 less than the width. What are the side lengths?

Independent Practice

Exit Ticket 1. Name the variables in the problem, then name any constraints that exist on the variables: John the plumber charges 15 dollars per hour in addition to a $50 fee. John’s work week consists of 40 hours. John’s insurance will not cover him past 40 hours of work per week. How many hours would he have to work if he is hoping to make $350?