4.1.2: Exercise 4.1

MAKING CONNECTIONS TO THE COLLABORATION

(1) Which of the following was one of the main mathematical ideas of the collaboration?

(i) Quantitative literacy is useful in comparing cell phone plans.

(ii) A horizontal line looks “flat.”

(iii) A linear model has a constant rate of change.

(iv) The only way to tell if a model is linear is to look at the graph.

DEVELOPING SKILLS AND UNDERSTANDING

The models you explored in Collaboration 4.1 are linear models. Linear models can represent a wide assortment of realistic problems, including problems that you may encounter in your personal life. Fortunately, linear models are simple to identify. Look for the following characteristics:

• Linear models are applicable when there is a constant rate of change between two variables.
• The graph of a linear model is a line.

• A linear model can be written in a format you may have seen in a previous course:
  y = mx + b where m is the slope and b is the vertical intercept or y-intercept. But note that linear models can also be written in other, equivalent formats such as Ax + By = C. One important characteristic is that the variables in the equation have the power of 1.

In Preparation 4.1, you learned about four ways to represent a linear model using the example of the model for the cost of driving Jenna’s car from Collaboration 3.2. The equation is shown below.

\(J = \dfrac{g}{18} + 0.0626\) where J is the cost per mile of driving and g is the price of gas per mile.

This is a linear model and can be written in the form y = mx + b. As you learned in Collaboration 1.4, a fraction is a way to write division. As you learned in Preparation 3.1, division can be written as multiplying by the reciprocal: \(\dfrac{g}{18} = \dfrac{1}{18}\cdot g\). The equation can be written as shown below.

\[J = \dfrac{1}{18}g + 0.0626\nonumber \]

The slope of this model is \(\dfrac{1}{18}\), or approximately 0.055, meaning the cost per mile of driving the car increases by $0.055 for each dollar that a gallon of gasoline costs. The vertical intercept (0.0626) represents the fixed cost of owning the car averaged out per mile. As you may recall from Collaboration 3.2, this was the cost of maintenance, insurance, and registration.

(2) Do the following representations depict a linear model or nonlinear model? Use the graph, equation, or table provided to determine your answer. Justify each decision based on one of the three characteristics listed above.

(a) From Exercise 3.6: \(R = \dfrac{t}{588.25} - 3.24\) where R is the percent of Americans who are retired in the year t. The graph is shown below.

(b) From Exercise 3.6: P = 120 + 2L, where P = the perimeter of a rectangular area (in feet), and
L = the length of the rectangle (in feet).

(c) From Collaboration 3.4: The graph for braking distance versus speed for the coefficient of friction of 0.8 and a roadway grade of 5% is shown below.

(d)

(e) The formula for the area of a circle: A = πr².

(3) Indicate if each of the following statements is true or false.

(a) The slope is a ratio that describes how one quantity changes (increases or decreases) when another quantity increases by 1 unit.

(i) True

(ii) False

(b) The slope of a linear model changes depending on the values substituted into the model.

(i) True

(ii) False

(c) The horizontal intercept is represented by an ordered pair in which the first value is zero: (0, __).

(i) True

(ii) False

(d) The vertical intercept is sometimes called a starting or initial value.

(i) True

(ii) False

(4) Sheila wants to lose weight to improve her health. She currently weighs 186 pounds and her goal is to weigh 140 pounds. After consulting with her doctor, she feels she can safely lose two pounds per week. The graph tracks the projected weight loss over time.

(a) Use the graph to estimate how many weeks it will take Sheila to achieve her desired weight goal.

(b) If Sheila’s next health screening is in four months, will Sheila have achieved her goal? Justify your answer.

(c) Write an equation for the weight loss trend. Use W = weight (lbs) and t = time (weeks).

(d) What is the slope of the equation you wrote in (c)? (Include the units in your response.)

(5) During the summer, Tamir likes to swim at his local pool a few times a week. He is debating if he should buy a “season pass” that allows him to swim as many times as he wants or just pay each time he goes swimming. The pool charges $4.50 each day if he pays each time. The “season pass” is $207, which is good for the entire summer (91 days).

(a) Fill in the following table to compare the two options.

(b) How many times would Tamir have to swim to make the “season pass” less expensive than paying for each visit individually?

At least ______ times

(c) Which of the graphs below models each situation? Assume each graph depicts Quadrant I only and that the origin is in the lower left hand corner of each graph. Assume number of days is on the horizontal axis and total cost is on the vertical axis.

Season pass: _______________ Paying per day: _______________

(6) Doctors use intravenous (IV) drips to deliver fluids and medications to patients. Suppose an IV drip starts with 1,000 mL of saline and dispenses fluid at a rate of 2.5 mL per minute.

(a) Write an equation for this situation using F to represent the amount of fluid (saline solution) left in the IV drip in milliliters and t to represent the time in minutes. (Hint: If you have trouble writing a model, try making a table first.)

(b) Is this a linear model? Explain your answer.

(c) How long will it take to use all of the saline solution? Give your answer in hours and minutes.

(d) Which of the following can be used to describe the point you found in (c)?

(i) Horizontal intercept

(ii) Vertical intercept

(iii) Slope

(iv) Origin

MAKING CONNECTIONS ACROSS THE COURSE

(7) The Bureau of Labor and Statistics published the following table on its website.¹

Employment in Healthcare by Industry Segment,
2021 and Projected Change, 2021–31 (employment in thousands)

(a) Based on this prediction, how many total healthcare diagnosing or treating practitioners will be employed in the United States in 2031?

(b) What is this number in words?

(c) On average, how many healthcare diagnosing or treating practitioners will be added each year during this period?

(d) Which of the following can be used to describe the value you found in (c)?

(i) Horizontal intercept

(ii) Vertical intercept

(iii) Slope

(iv) Origin

(e) Write a linear equation for this relationship. Let W = the number of workers, and t = the number of years after 2021 (for example, t = 1 for 2022).

____________________________________________

¹ https://www.bls.gov/emp/tables/occupational-projections-and-characteristics.htm