4.7.2: Exercise 4.7

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    MAKING CONNECTIONS TO THE COLLABORATION

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

    (i) Exponential growth starts slowly, but then speeds up. Linear growth is constant.

    (ii) Compound interest has a special formula because it is not the same as exponential growth.

    (iii) “Compounded weekly” means that the interest will be compounded 52 times in a year.

    (iv) Payday loans are not a good way to borrow money because they have very high interest rates.

    (2) The graph below is an exponential model of the levels of toxic waste in a lake. The amount of waste is decreasing at a rate of 12% each year.

    Line graph showing the levels of toxic waste in Lake Meadows. Y-axis represents the amount of toxic waste in pounds, ranging from 0 to 1,200. X-axis represents the number of years, ranging from 0 to 14. 0 = 1,000 2 = in between 600 and 800 4 = 600 6 = in between 400 and 600 8 = in between 200 and 400 10 = in between 200 and 400 12 = 200

    (a) If the trend continues, which of the following statements would be correct?

    (i) The decrease in the level of waste from Year 12 to Year 14 will be the same as the decrease from Year 10 to 12.

    (ii) The decrease in the level of waste from Year 12 to Year 14 will be less than the decrease from Year 10 to 12.

    (iii) The decrease in the level of waste from Year 12 to Year 14 will be more than the decrease from Year 10 to 12.

    (b) Would your answer in (a) change if this had been a linear model? Explain.

    DEVELOPING SKILLS AND UNDERSTANDING

    (3) Match the following descriptions with the graphs (on the next page). Each description may match more than one graph. Some graphs may match more than one situation and some graphs may not match any situation.

    (a) This graph represents the monthly cost of a gym membership plan. Its monthly fee is $20 with an additional charge of $5 per class, where Y = monthly cost of gym membership and t = number of classes.

    (b) This graph represents the monthly cost for a gym membership that has a fixed monthly charge of $20 with free unlimited classes where Y = monthly cost of gym membership and t = number of classes.

    (c) This graph represents the amount in an account that decreases by $5 each month where Y = balance in the account and t = months.

    (d) This graph represents the amount in an account that decreases by 25% each month where Y = balance in the account and t = months.

    (e) This is a graph of Y = −5t + 40

    (f) This is a graph of Y = 30(1.25)t

    (g) This is a graph of an exponential function.

    (h) This is a graph of a linear function.

    I.

    Graph showing an increasing curb, indicating the following: Start: 0 = 30 End: 5 = just above 90

    II.

    Graph showing a decreasing straight-line, indicating the following: Start: 0 = 40 End: 5 = in between 10 and 20

    III.

    Graph showing a straight-line, indicating the following: Start: 0 = 20 End: 5 = 20

    IV.

    Graph showing a decreasing curb, indicating the following: Start: 0 = 50 End: 5 = in between 10 and 20

    V.

    Graph showing an increasing squiggly line, indicating the following: Start: 0 = 30 End: 5 = in between 40 and 50

    VI.

    Graph showing an increasing straight-line, indicating the following: Start: 0 = 20 End: 5 = 45

    (4) In a previous collaboration, you saw that a person’s Blood Alcohol Content (BAC) after t hours could be approximated by Widmark’s equation. Paula is part of a research study to verify Widmark’s equation. She weighs 130 pounds. The researchers ask her to drink three beers. After one hour, her BAC is at 0.10. After three hours her BAC is at 0.08.

    (a) Find an equation of the line using her BAC measurements where B is her BAC after t hours. Hint: The time and BAC can be written as ordered pairs: (1, 0.10) and (3, 0.08). It might help to graph the two points and draw the line between them.

    (b) What is the slope of your equation and how would you interpret this value in the context of the situation?

    (c) What are the horizontal and vertical intercepts of your equation and how would you interpret these values in this scenario?

    (d) Recall Widmark’s equation:

    \[B = -0.015\cdot t + \left( \dfrac{2.84\cdot N}{W\cdot g}\right)\nonumber \]

    where N is the number of “standard drinks,” W is his/her weight in pounds, and g = gender constant (0.68 for men and 0.55 for women). One standard drink is a 12-ounce beer, 5-ounce glass of wine, or 1.5-ounce shot of liquor.

    (i) Simplify this equation for Paula’s case, leaving t as a variable.

    (ii) State the slope and vertical intercept for this simplified equation, and interpret these values.

    (e) Looking at the models you created in (a) and (d), compare the slopes of the two equations. Which model has the steepest slope? What does that mean in this context?

    (5) Caffeine is another chemical in food and beverages. However, caffeine is processed in the body differently than alcohol. For most people, five hours after eating or drinking something with caffeine only about half of the caffeine is left in the bloodstream. In another five hours (10 hours total) half of that is left, and so on. This situation is exponential because the amount of caffeine decreases by 50% every five hours.20

    (a) Suppose Fred drinks a tall Café Americano from Starbucks, which has about 150 mg of caffeine.21 Complete the table below for this situation. Round to the nearest tenth. The first two entries are done for you.

    Time in Hours

    Amount of Caffeine Left in His Body (mg)

    0

    150

    5

    75

    10

    (i)

    15

    (ii)

    20

    (iii)

    25

    (iv)

    30

    (v)

    (b) Plot the data from (a) on a graph.

    (c) Suppose his wife, Wilma, drinks a 12-oz Diet Coke, which has 45 mg of caffeine. However, Wilma is using an oral contraceptive. When someone is on “the pill,” it takes about 10 hours (twice as long) for half of the caffeine to leave the bloodstream, and another 10 hours for half of the remaining to leave, etc. Complete the table below for this situation. Round to the nearest tenth. Then plot the amount of caffeine left in Wilma’s body on the same graph as in (b).

    Time in Hours

    Amount of Caffeine Left in Her Body (mg)

    0

    (i)

    10

    (ii)

    20

    (iii)

    30

    (iv)

    (d) Looking at your graph of Fred and Wilma’s data, who has:

    (i) more caffeine in their body in the first few hours?

    (ii) more caffeine in their body in the long term?

    (e) Based on your graphical models, is there a time that either Fred or Wilma will have no caffeine left in their body? Explain your reasoning.

    MAKING CONNECTIONS ACROSS THE COURSE

    (6) In Exercise 4.5, you modeled the world population based on the growth rate in 2023. To plan for future needs, it is necessary to project how many people there will be in a certain place, but it is also necessary to know information about the ages of the people. This type of information is often displayed in a type of graph called a population pyramid. The population pyramid shown below is from the CIA World Factbook. Look at it carefully and think about what information is shown before continuing.22

    United States - 2023

    Horizontal bar graph showing the population in millions by age group in the United States in 2023. Y-axis represents the age groups, ranging from 0-4 to over 100. X-axis represents the population in millions, ranging from 0 to 15. Male age groups 0-4, 5-9, 10-14, 15-19, 20-24, 25-29, 30-34, 35-39, 40-44, 45-49, 50-54, 55-59, 60-64, all have a population between 10 and 12.5. Female age groups 0-4, 5-9, 10-14 all have a population around 10. Female age groups 15-19, 20-24, 25-29, 30-34, 35-39, 40-44, 50-54, 55-59, 60-64, 65-69 all have a population between 10 and 12.5. 45-49 = around 10. Male 65-69 = between 7.5 and 10. Female 65-69 = just over 10. Male 70-74 = just under 7.5. Female 70-74 = in between 7.5 and 10. Male 75-79 = just over 5. Female 75-79 = in between 5 and 7.5. Male 80-84 = in between 2.5 and 5. Female 80-84 = close to 5. Male 85-89 = between 0 and 2.5. Female 85-89 = around 2.5. Male 90-94 = about 1. Female 90-94 = about 2.5. Male 95-99 and 100+ = very close to 0. Female 95-99 = about 1. Female 100+ = very close to 0.

    This population pyramid is for the U.S. in 2023. The left side of the pyramid represents males; the right represents females. Each bar represents the age group listed in the center column. This pyramid measures the population in millions along the horizontal axis. Sometimes these types of graphs measure the population in percentages instead of numbers. Use this graph to answer the following questions.

    (a) Based on the graph, which of the following statements is the most accurate?

    (i) In the population under 30 years old, the number of men and the number of women are similar.

    (ii) In the population over 79 years old, the number of men and the number of women are equal.

    (iii) There are fewer people in the age range of 35–44 than in the range of 45–55.

    The graph below illustrates Mexico’s population in 2023.23

    Mexico - 2023

    Horizontal bar graph showing the population in millions by age group in Mexico in 2014. Y-axis represents the age groups, ranging from 0-4 to over 100. X-axis represents the population in millions, ranging from 0 to 8. 0-4: Male = in between 4 and 6; Female = in between 4 and 6. 5-9: Male = in between 4 and 6; Female = in between 4 and 6. 10-14: Male = in between 4 and 6; Female = in between 4 and 6. 15-19: Male = in between 4 and 6; Female = in between 4 and 6. 20-24: Male = in between 4 and 6; Female = in between 4 and 6. 25-29: Male = in between 4 and 6; Female = in between 4 and 6. 30-34: Male = in between 4 and 6; Female = in between 4 and 6. 35-39: Male = in between 4 and 6; Female = in between 4 and 6. 40-44: Male = in between 4 and 6; Female = in between 4 and 6. 45-49: Male = in between 2 and 4; Female = in between 4 and 6. 50-54: Male = in between 2 and 4; Female = around 4. 55-59: Male = in between 2 and 4; Female = in between 2 and 4. 60-64: Male = in between 2 and 4; Female = in between 2 and 4. 65-69: Male = in between 0 and 2; Female = about 2. 70-74: Male = about 1; Female = about 1.5. 75-79: Male = under 1; Female = about 1. 80-84: Male = about 0.5; Female = just over 0.5. 85-89: Male = close to 0; Female = just under 0.5. 90-94: Male = just over 0; Female = just over 0. 95-99: Male = just over 0; Female = just over 0. Over 100: Male = just over 0; Female = just over 0.

    (b) Indicate if each statement applies to the United States, Mexico, or both.

    (i) Beyond age 65 there are consistently more women than men in each category.

    (ii) The number of people 0–9 years old is about the same as the number of people 10–19 years old.

    _____________________________________________

    20 http://en.Wikipedia.org/wiki/Caffeine

    21 https://www.starbucks.com/menu/product/406/hot/nutrition

    22 https://www.cia.gov/the-world-factbook/countries/united-states/

    23 https://www.cia.gov/the-world-factbook/countries/mexico/

    This page titled 4.7.2: Exercise 4.7 was last modified on Wed, 15 May 2024 07:39:05 GMT and is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Carnegie Math Pathways (WestEd) .