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Healthcare Scenario–Unit Conversions, Dimensional Analysis, Body Mass Index and Weight Loss
Prepared by: Pete Kaslik
Healthcare Contributor: Terry Tatko, Franciscan Healthcare
Introduction to Math Teachers
This scenario will:
Introduce students to mathematical applications in the healthcare, fitness, and nutrition fields
Increase familiarity with metric system
Show unit conversion with dimensional analysis
Give practice in substituting into algebraic equations
Give practice in solving algebraic equations
GLEs (Math Focus Group, March 26, 2007)
1.1.11.1.21.1.31.1.41.1.61.1.81.2.31.5.11.5.21.5.41.5.62.1.1 2.1.2 2.1.3 2.2.1 2.2.2 2.2.4 3.2.2 3.3.1 3.3.2 4.2.1 5.3.1 5.3.2
College Readiness Standards (Math Focus Group, March 26, 2007)
1.11.21.3 2.12.22.33.13.23.34.14.24.3 7.1 7.3 8.1 8.3
Introduction
The household where Jordan grew up had an endless supply of pastries, snack foods, white bread, ice cream and candy. In fact, candy was a frequent reward for him as a child. If he cleaned his room, he was rewarded with candy. If he didn’t fight with his brother, more candy. In school, if he turned in an assignment, the teacher gave him candy. By the time he was a senior in high school, he was unmotivated and out of shape. During a scheduled physical at FranciscanHospital in Tacoma, the doctor examining him said little until the physical was complete. Then he sat down across from Jordan, looked him straight in the eye and told him he would not make it to age thirty. Jordan had a Body Mass Index of 40.7 kg/m2, elevated blood pressure of 138/92, a resting pulse rate of 79 and early signs of diabetes. His apathy and lack of motivation were also related to his poor health.
While he had long since become used to people teasing him about his weight and generally making fun of him, the matter-of-fact, uncompromising tone of the doctor struck him in a way that no one else had. Jordan looked in a mirror. He didn’t like what he saw. He knew the doctor was right and although the doctor’s words stung his ego, he knew he had to do something.
More and more people in the United States, including young people, are overweight or obese. Obesity is associated with a great number of serious medical conditions that threaten life, and cause a decrease in the quality of life for those who suffer them.
The Health Care Industry
A great number of professionals are dedicated to addressing the needs of those who are sick, as well as those who wish to improve their quality of life or increase their overall health. In addition to doctors, nurses, medical technicians, and others who work in hospitals and clinics, there are many professions in both clinical and non-clinical settings that help people become or stay healthy, including dieticians, nutritionists, personal trainers, and fitness coaches.
Healthcare is a high growth industry representing approximately 15% of the gross domestic product (GDP) in the United States. Healthcare jobs range in salaries according to degree of skill and experience. Medical personnel shortages are critical, with the average age of most healthcare workers over 45. The industry is increasingly reliant on technology and a workforce that is skilled at working effectively with people in a high tech environment. Skilled healthcare workers can earn a good salary, and generally have a wide choice of geographic areas and work settings where they can find employment.
The doctor told Jordan about Franciscan’s weight loss program in which fitness coaches and nutritionists work together to help participants to reach their goals.
As an intern, you have been assigned to help Jordan and others with the calculations necessary to set goals and monitor progress.
Part 1. Body Mass Index
The fitness coach explained the body mass index to the group. Body mass index (BMI) is the ratio of mass, in kilograms to the height, in meters squared, . It has units of . The BMI is used to calculate a person’s weight relative to his/her height in order to compare it to an acceptable standard, but it is only an approximate assessment. For example, muscular individuals (mesomorphic) tend to have inflated BMI values. The general breakdown of BMI values[1] is
- Starvation: less than 15
- Anoretic(n.)/Anorexic(adj.): less than 17.5
- Underweight: less than 18.5
- Ideal: from 18.5 to 25
- Overweight: from 25 to 30
- Obese: from 30 to 40
- Morbidly Obese: greater than 40
Jordan is 5’11 and weighs 290 lbs. To help him calculate his BMI, it is necessary to convert his weight to kilograms and height to meters.
With this BMI, Jordan was considered morbidly obese.
Help Jordan’s classmates calculate their BMI.
1. Cindy is 5’3” and weighs 215 pounds.
2. Alex is 6’2” and weighs 238 pounds.
3. Sandra is 5’6” and weighs 196 pounds.
Part 2. Finding a Good Weight
The goal for improving one’s health is to have a reasonable weight, but not one that is so low it causes stress to maintain. Jordan sets his BMI goal at 25. He can use this goal to help calculate his desired weight. We will represent weight with the variable W.
To solve for W, multiply both sides of the equation by 1.82.
81 = W. Thus, his goal weight is 81 kilograms. This can be converted to pounds using dimensional analysis.
For Jordan, the objective was clear. He needed to lose 112 pounds, which is over 38% of his weight. This can be calculated by .
Now help Jordan’s classmates calculate their goal weight and percent of body weight they must lose.
1. Cindy would like a BMI of 28.
2. Alex would like a BMI of 22.
3. Sandra would like a BMI of 24.
Part 3. Calories
To lose weight, it is necessary to expend more calories than are consumed. Calories are a measure of energy, thus food calories represent the energy stored in food. Calories burned during exercise represent the energy expended doing the exercise. If more calories are consumed than expended, a person will gain weight. If more calories are expended than consumed, a person will lose weight. 3,500 excess calories create one pound of fat, which is the body’s way of storing energy[2]. Expending 3,500 more calories than are consumed removes one pound of fat. The amount of calories used for simple existence is called the basal metabolic rate. Additional calories are used for any actions beyond this minimal existence. In particular, exercise is an excellent way to increase caloric output.
The basal metabolic rate (BMR) is calculated using the formulas below:
The original equations from Harris and Benedict are:
- for men, 66.4730 + (13.7516 * w) + (5.0033 * s) − (6.7550 * a)
- for women, 655.0955 + (9.5634 * w) + (1.8496 * s) − (4.6756 * a) >
where w = weight in kilograms, s = stature in centimeters, and a = age in years[3]
Once the BMR is known and the amount of exercise is determined, the daily caloric intake can be calculated.
The BMR formula requires a weight in kilograms, which is already known for Jordan. The height is in centimeters, which is the height in meters times 100. Jordan is 18 years old. Jordan’s height in centimeters is. Substitute into the formula for men. BMR = 66.4730 + (13.7516 * w) + (5.0033 * s) − (6.7550 * a)
BMR = 66.4730 + (13.7516 * 81) + (5.0033 * 180) − (6.7550 * 18)
BMR = 1959 calories
Since this value is dependent upon weight, it will need to be recalculated periodically as Jordan loses weight.
Help Jordan’s classmates find their BMR.
1. Cindy is a 30 year old female.
2. Alex is a 37 year old male.
3. Sandra is a 42 year old female.
Besides the calories required for simple existence, the primary source of caloric expenditure will come from an exercise program. Jordan was given several options for an exercise program. He decides he would like to learn rowing, since he lives in the Puget Sound region. Rowing is one of the best sports for burning calories and improving cardiovascular fitness. It has the added benefit of being non-weight bearing. Also, rowing clubs exist in the region, providing competitions for all ability levels. It is estimated that rowers burn over 700 calories per hour.
Before calculating his caloric intake, Jordan must determine his caloric output. His goal is to consume about 500 calories less per day then he expends. That way, in the course of one week, he would use 3500 more calories then he consumes, resulting in the loss of one pound per week, an excellent rate for losing weight.
The amount of calories Jordan burns each day is equal to his basal metabolic rate plus those used in exercise. Thus Calories burned = 1959 + 700h, where h represents hours of rowing. This equation fits the form of the word equation:
calories burned = BMR + Exercise calories per hour · number of hours
The algebraic equation can be written C=B + Eh. The variable h is the independent variable in this linear equation. The variable C is the dependent variable. B represents the basal metabolic rate and E is the rate of caloric usage during exercise. C will depend upon the number of hours of exercise that are done.
The calories that are burned each day can be determined by deciding on the time (in hours) to be spent on the exercise, substituting the value into the equation for h, and simplifying. It is important not to start with too much exercise, so assume Jordan starts with 15 minutes, or 0.25 hour per day. Then C = 1959 + 700(0.25). Simplifying gives C = 2134 calories. If he eats 500 fewer calories than he burns, then his daily intake of calories will be 1634.
Create an equation for each of Jordan’s classmates that can be used to compute their daily caloric expenditure.
1. Cindy’s chosen exercise burns 350 calories/hour.
2. Alex’s chosen exercise burns 500 calories/hour.
3. Sandra’s chosen exercise burns 425 calories/hour.
Use the formulas and the time spent exercising to determine the number of calories burned per day.
1. Cindy will exercise for 0.5 hours/day.
2. Alex will exercise for 0.75 hours/day.
3. Sandra will exercise for 0.35 hours/day.
If the goal is to consume 500 fewer calories than are burned, how many calories can Cindy, Alex and Sandra consume?
Part 4. What to Eat
Recommendations vary for the distribution of calorie among the three essential food groups: proteins, fats and carbohydrates. Assume Jordan plans for his caloric intake to be in the proportion of 15% proteins, 30% fat and 55% carbohydrates. The energy yield per gram is as follows: Carbohydrate 4 calories, Fats 9 calories and Protein 4 calories. This information can be used to determine the number of grams of proteins, fats and carbohydrates Jordan should eat each day.
Now that you have seen how Jordan calculated his caloric intake, help the others in the class.
1. Cindy wants a diet that is 20% protein, 10% fat, 70% carbohydrates
2. Alex wants a diet that is 25% protein, 30% fat, 45% carbohydrates
3. Sandra wants a diet that is 15% protein, 25% fat, 60% carbohydrates
Just before class ends, Antonio arrives and you need to help him determine the amount of proteins, fats and carbohydrates he should be consuming daily. The necessary information you need to determine this is provided in the table.
Part 5. Review
Name: AntonioAge: 28Sex: Male
Height: 6’4Weight: 346 lbs
BMI Goal: 23Exercise Calories: 625 calories/hour
Nutrient distribution: 40% protein, 35% fat, 25% carbohydrates
Find:
BMI:
% of body weight to lose
BMR
Equation for calculating Calories burned per day
Calories to be consumed per day
Grams of protein per day
Grams of fat per day
Grams of carbohydrates per day
[1] (
[2] (
[3] (