2.1.2: Exercise M.1
- Page ID
- 148550
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)(1) Jenna is using her car for work and created a model for the cost, per mile, of driving her car. It is given by the equation:
\[ J = \dfrac{g}{22} + 0.0536 \nonumber \]
where J = Cost, per mile, of driving Jenna’s car (in dollars)
g = Cost of gas (dollars/gallon)
(a) If the cost of gas is $3.92 per gallon, what is the cost, per mile, of driving Jenna’s car? Round to the nearest cent (e.g., $0.87).
(b) Is g a variable? Explain your reasoning.
(c) What other variables (in addition to cost of gas) could have been included for this context?
(d) Let’s think about one variable: the cost of insurance. Describe one way to measure that variable.
(2) In baseball statistics, slugging percentage (abbreviated SLG) is a popular measure of the power of a batter. It is calculated as total bases earned divided by at-bats:
\[ SLG = \dfrac{S + 2*D + 3*T + 4*HR}{AB} \nonumber \]
where AB is the number of at-bats for a given player, and S, D, T, and HR are the number of singles, doubles, triples, and home runs, respectively. (Walks are specifically excluded from this model and do not count as at-bats.)
(a) What are the units for slugging percentage?
(b) The table below shows statistics for the New York Yankees baseball player Aaron Judge.5 Fill in Judge’s missing SLG for the years 2018, 2020, and 2022. Round your answers to three decimal places. In which year did Judge have the most power?
| Year | AB | S | 2B | 3B | HR | SLG |
| 2017 | 542 | 75 | 24 | 3 | 52 | 0.627 |
| 2018 | 413 | 66 | 22 | 0 | 27 | |
| 2019 | 378 | 57 | 18 | 1 | 27 | 0.540 |
| 2020 | 101 | 14 | 3 | 0 | 9 | |
| 2021 | 550 | 95 | 24 | 0 | 39 | 0.544 |
| 2022 | 570 | 87 | 28 | 0 | 62 |
(c) List two ways a player could increase his or her power rating.
(3) In Collaboration M.1, we used BMI to determine whether athletes were obese. The BMI calculation is based on the relationship between weight and height. There are other formulas that try to account for additional variables when estimating body fat. Deurenberg and Weststrate6 developed the following formula for estimating body fat percentages:
- Men’s Adult Body Fat Percentage = 1.2*BMI + 0.23*Age – 16.2
- Women’s Adult Body Fat Percentage = 1.2*BMI + 0.23*Age – 5.4
The table below provides the names and ages (at the time their height and weight were measured) of the athletes you classified in class.
| Athlete | Age |
BMI (from Question 6) |
BMI Classification (from Question 7) |
Estimated Body Fat Percentage (BF%) | BF% Classification |
| Rory McIlroy | 27 | ||||
| Julianna Pena | 33 | ||||
| Cam Gallagher | 30 | ||||
| Breanna Stewart | 28 | ||||
| Shaquille O’Neal | 36 | ||||
| Maria Sharapova | 29 |
Fill in the BMI and BMI classifications from Questions 6 and 7. Then complete the next two questions.
- Use Deurenberg and Weststrate’s formula to estimate each athlete’s body fat percentage at the age given. Enter your answers in the “Estimated Body Fat Percentage” column.
(b) The American Council on Exercise uses the following classification system based on Body Fat Percentage:7
| Description | Women | Men |
| Essential Fat | 10–13.99% | 2–5.99% |
| Athletes | 14–20.99% | 6–13.99% |
| Fitness | 21–24.99% | 14–17.99% |
| Average | 25–31.99% | 18–24.99% |
| Obese | 32% + | 25% + |
Find the athletes’ classifications based on their estimated body fat percentage and enter the information in the “BF% Classification” column of the table above. (Note: In the table above, use the middle column for Julianna Pena, Breanna Stewart, and Maria Sharapova. Use the third column for Rory McIlroy, Cam Gallagher, and Shaquille O’Neal.)
(c) Compare the classification based on BMI for Breanna Stewart to the classification based on body fat percentage. Do the classifications match? What would it mean if the classification did not match?
(4) Do you think that BMI is a reasonable estimate of body fat composition? Why or why not? Does BMI help classify people as “overweight?” Is it useful for classifying all kinds of people, at all weight levels? What good does using the BMI formula do? What harm does using the BMI formula do?
(5) Recall Deurenberg and Weststrate’s formula for estimating body fat percentages:
- Men’s Adult Body Fat % = 1.2*BMI + 0.23*Age – 16.2
- Women’s Adult Body Fat % = 1.2*BMI + 0.23*Age – 5.4
Using these formulas, consider how the two different mathematical models work in the following scenarios.
(a) If Pedro keeps his BMI constant over time, what will happen to his estimated body fat percentage as he ages?
(i) The body fat percentage will increase.
(ii) The body fat percentage will decrease.
(iii) The body fat percentage will stay the same.
(b) Aimee and Christopher are twin sister and brother, respectively. If they have the same height and weight, who will have a greater estimated body fat percentage?
(i) Aimee.
(ii) Christopher.
(iii) They will have the same body fat percentage.
(c) This year, Nisheidi increased her BMI by 1 from last year. She also got one year older. Which increase, age or BMI, will have a greater impact on her estimated body fat percentage?
(i) Increased age by 1 year.
(ii) Increased BMI by 1 point.
(iii) They would have the same impact.
(6) Many pediatric dosages of drugs are determined by using the child’s body surface area. The Mosteller formula is often used for this purpose.8 To determine a child’s body surface area (in square meters), the child’s height (in inches) and weight (in pounds) are multiplied together; this number is divided by 3131 and then a square root is taken of the result. Thus
\[ A = \sqrt{\dfrac{h * w}{3131}} \nonumber \]
where:
A = the child’s body surface area (in square meters)
h = the child’s height (in inches)
w = the child’s weight (in pounds)
(a) Use this formula to find the surface area of a child who is 4 feet tall and weighs 75 pounds. Round to the nearest hundredth of a square meter. (Note: You will need to first convert the height into inches.)
(b) To get the dosage of an allergy medicine for a child, the following formula is used:
\[ D = \dfrac{25 * A}{1.73}mg \nonumber\]
where A is the body surface area, in square meters. Calculate the dosage of this allergy medicine for this child. Round to the nearest tenth of a milligram.
(c) One practitioner forgot to convert the feet to inches and uses “4” in the equation for height. What surface area did this practitioner calculate? Round to the nearest hundredth of a square meter.
(d) What was the resulting dose from making this error? Round to the nearest tenth of a milligram.
(e) What do you think would be the result of the miscalculated prescription?
(7) In Collaboration M.1, you saw that mathematical models can be represented by equations containing variables. Equations are just one of the many types of representations that can be used to describe a mathematical model. In addition to equations, many mathematical models are represented by tables or graphs. One such graphical model is a pediatric growth chart.
Pediatric growth charts have been used by pediatricians, nurses, and parents to track the growth of infants, children, and adolescents in the United States since 1977. These growth charts consist of a series of percentile curves that illustrate the distribution of various measurements in relation to age and gender, including height, weight, head circumference, and Body Mass Index.
One of the newer charts is the Body Mass Index-For-Age (BMI-For-Age) chart for boys ages 2 to 20 years, shown in the figure below. This chart can be used to determine the percentile rank for a child, given the age and the BMI of the child.
For example, Enrique is 8 years old and he has a BMI of 17.5. According to the BMI-For-Age chart below, Enrique falls into the 75th percentile (less than the 85th percentile). This means that 75% of all 8-year-old boys have a BMI that is less than Enrique’s, or only 25% of all 8-year-old boys have a BMI that is greater than Enrique’s. Look at the chart, and make sure you understand how it was used to determine that Enrique falls in the 75th percentile, given his age and BMI.
Use the chart to answer the following questions:
(a) Alex is a 12-year-old boy who weighs 120 pounds and measures 4 feet 11 inches tall. Calculate Alex’s BMI using the formula from the in-class lesson
\[ BMI = \dfrac{weight\;(in\;kilograms)}{[height\;(in\;meters)]^2} \nonumber \]
Then use the chart above to determine his percentile rank.
(b) Joshua is 16 years old and falls into the 25th percentile. What is his approximate BMI?
(c) A boy is tracked from age 6 until age 15 using this BMI-For-Age growth chart. Suppose that throughout this entire time, he has always been in the 85th percentile rank. What happens to his BMI as he ages from 6 to 15?
(d) A boy is tracked from age 6 until age 15 using this BMI-For-Age growth chart. Suppose that throughout this entire time, his BMI has stayed at 17 kg/m2. What happens to his percentile rank as he ages from 6 to 15?
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5 https://www.baseball-reference.com/players/j/judgeaa01.shtml
6 http://www.halls.md/bmi/fat.htm
7 http://www.acefitness.org/blog/112/what-are-the-guidelines-for-percentage-of-body-fat/
8 R.D. Mosteller, “Simplified Calculation of Body Surface Area,” New England Journal of Medicine 317, no. 17 (1987): 1098.