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Math Lecture Notes
Health

In-Class Activity link to here link to index

Breathing or Heart Rate?

Extra Energy, Extra Oxygen

Breathing or Heart Rate

Calories are a measure of energy. Our body uses energy even when resting for breathing, blood circulation, maintenance of body temperature and growing and repairing cells. Exercising needs more energy.

How can we find the number of calories burned when exercising?

This pattern might be complicated. It surely depends upon the type of exercise, how intensely it is done, and for how long the person exercises. It could also sensibly depend upon the person's age, weight, or sex.

As an in-class activity, let's just focus on measuring one person doing one type of exercise.

Get Some Measurements

When we exercise our muscles need more oxygen. For aerobic exercise, the amount of extra oxygen needed is proportional to the amount of energy used exercising. To get that extra oxygen to our muscles, our breathing and heart rate increase.

1. Pick an exercise whose intensity you can change accurately. Record your breathing and heart rates at different intensities.

Two Possible Exercises

As an example, you might walk through the hallways for a minute to see what distance you go. Then jog that distance twice in a minute. Then run that distance three times in a minute.

As another example, you might do liesurely jumping jacks for a minute. Then do twice as many in a minute. Then do three times as many in a minute.

2. Look for a pattern. Does breathing rate or heart rate better estimate exercise energy use?

The simplest result would be if either your breathing rate or heart rate (or both) doubled and tripled when the exercise intenisty doubles and triples.

Maybe the pattern is not that simple. Maybe either breathing or heart rate will increase more or less than doubling or tripling when exercise intensity doubles or triples.

Measured or Extra?

3. Try subtracting your baseline breathing and heart rates.

If neither your breathing or heart rate (or both) make a good pattern, perhaps your resting metabolism is the culprit. After all, the exercise was needing extra oxygen, so perhaps what we should look at is the extra breathing and heart rates?

If you subtract the resting rates it will isolate the increases caused by exercise. Does a better pattern happen if you subtracting your resting breathing and heart rates from the exercise values at all intensities?

Facts link to here link to index

Calories

Calories in a Gram

Not all calories are created equal! One gram of fat has more than twice the calories of a gram of carbohydrate or protein.

carbohydrate protein fat

Calories per Gram

      • Carbohydrates have 4 calories per gram

      • Proteins have 4 calories per gram

      • Fats have 9 calories per gram

Those are measurement conversion rates. They let us switch between grams and calories.

bagel nutrition information

4. Arthur is in a rush this morning and eats a plain 4" bagel in the car for his breakfast. How many of the bagel's calories are from carbohydrates? from proteins? from sugar?

4. The bagel has 47 × 4 = 188 calories from carbohydrates.

It has 9 × 4 = 36 calories from protein.

Sugar is a kind of carbohydrate, so it also has 5 × 4 = 20 calories from sugar.

5. The bagel package says the bagel has 1 gram of fat, which should be 9 calories. But the package also says the bagel has 13 calories from fat. What is the most likely explanation for this difference?

5. The bagel probably does have 13 calories from fat. This would be 13 ÷ 9 ≈ 1.4 grams of fat. The manufacturer rounded down to "1 gram" for the label.

6. One serving of Lite Chocolate Frosted Sugar Bombs has 1.3 grams of fat, 22 grams of carbohydrates (including 9 from sugar), and 2.5 grams of protein. Change to calories these amounts of fat, carbohydrate, sugar, and protein.

6. The cereal has 1.3 × 9 ≈ 12 calories from fat.

Its has 22 × 4 = 88 calories from carbohydrates.

As part of those carbohydrates, 9 × 4 = 36 calories are from sugar.

It has 2.5 × 4 = 10 calories from protein.

(By the way, the nutrition information for this fake cereal is actually from a popular "healthy" cereal!)

Calories in a Day

The USDA has recommendations for what someone on a 2,000 calorie per day diet should be eating. (Labels on food packages refer to this information.)

2,000 Calories per day Diet

• 300 grams of carbohydrate per day (1,200 calories)
• at least 53 grams of protein per day (212 calories or more)
• less than 65 grams of fat per day (585 calories or less)

Of these amounts, at least 25 of the carbohydrate grams should be dietary fiber, and no more than 20 of the fat grams should be saturated fat.

What about a person who is more active or larger and eats more? The recommendations for a 2,500 calorie per day diet have slightly higher amounts.

2,500 Calories per day Diet

• 375 grams of carbohydrate per day (1,500 calories)
• at least 70 grams of protein per day (280 calories or more)
• less than 80 grams of fat per day (720 calories or less)

Of these amounts, at least 30 of the carbohydrate grams should be dietary fiber, and no more than 25 of the fat grams should be saturated fat.

Most people do not fit either set of guidelines exactly.

7. Arthur should be eating 2,250 calories per day. What are his USDA recommendations for grams per day of carbohydrates, protein, and fat?

7. Arthur is halway between the 2,000 Calories Per Day guidelines and the 2,500 Calories Per Day guideliness. So we should average those.

For carbohydrates: (300 + 375) ÷ 2 ≈ 338 grams per day.

For protein: (53 + 70) ÷ 2 ≈ 62 grams per day.

For carbohydrates: (65 + 80) ÷ 2 ≈ 73 grams per day.

Burning Calories

Activity
 
Calories Per Pound
Per Minute
ballroom dancing0.023
grocery shopping0.028
walking0.037
weight lifting0.039
bicycling0.045
aerobic dancing0.061
basketball0.063
swimming0.070
running0.090

How can we find the number of calories burned when exercising?

This question is easy to answer because burning calories only depends upon the weight of the person exercising and how long they exercise. Unlike the formulas we already learned for BMR and DCI, height and age do not matter for calculating calories burned.

The chart on the right that estimates, for different activities, how many calories are burned per minute per pound of the person exercising.

If you are curious about other activities, more charts such as this one are easy to find in books and on the internet. But their information should always be taken with some skepticism. Two different people can participate in the same activity with very different levels of physical exertion. For example, this chart does not differentiate between bicycling at a lazy pace to the corner store or rushing over the Donald Street mountain pass as part of a cycling race.

8. Arthur weighs 150 pounds. He runs for an hour and then does a half hour of weight lifting. How many calories does he burn?

8. For Arthur's running we multiply:
155 pounds × 60 minutes × 0.09 calories per pound per minute ≈ 810 calories.

For Arthur's weight lifting we multiply:
150 pounds × 30 minutes × 0.039 calories per pound per minute ≈ 176 calories.

So Arthur's total is about 810 + 181 = 986 calories.

9. Odette weighs 110 pounds. She bicycles to the gym in 10 minutes, plays basketball for two hours, and then bicycles home in 20 minutes. How many calories does she burn?

9. For Odette's bicycling (30 minutes total) we multiply:
110 pounds × 30 minutes × 00.04509 calories per pound per minute ≈ 149 calories.

For Odette's basketball we multiply:
110 pounds × 120 minutes × 0.063 calories per pound per minute ≈ 832 calories.

So Odette's total is about 149 + 832 = 981 calories.

10. Odette weighs twice as much as her daughter. If they both do the same exercise, does Odette always burn twice as many calories?

10. In theory, yes, doubling the weight will double the answer. But in real life it seldom happens that an adult and child will exercise in exactly the same way for a long length of time. Perhaps Brianna's daughter is unusual, and when they go on walks will stay by her mother's side instead of frequently running ahead, or falling behind and then running to catch up.

Fat vs. Carbohydrates Burned

Another important aspect to the issue of how many calories are burned by different activities is what kind of calories are burned. A person wanting to lose weight desires to burn as many fat calories as possible, not only carbohydrate calories.

The practical lesson is simply "never mind and exercise a lot." But this is a math class, so we will make the issue complicated.

The trickiness comes from two effects working against each other.

When you do a more intense exercise then your metabolism burns a smaller percentage of fat calories. This happens because busy muscles need carbohydrate calories for quick energy and health.

But a more intense exercise burns many more calories total! The total number of fat calories burned still comes out ahead.

11. When walking, about three-quarters of the calories burned will be fat calories and one-quarter will be carbohydrate calories. When running, about half of the calories burned will be fat calories and half carbohydrate calories. Consider two people who both weigh 150 pounds who both exercise for 30 minutes: one person walks and the other person runs. How many fat and carbohydrate calories do each burn?

11. For the walker's total calories we multiply:
150 pounds × 30 minutes × 0.037 calories per pound per minute ≈ 167 total calories burned.

75% of those 167 calories are fat:
0.75 × 167 calories ≈ 125 fat calories.

25% of those 167 calories are carbohydrate:
0.25 × 167 calories ≈ 42 carbohydrate calories.


For the runner's total calories we multiply:
150 pounds × 30 minutes × 0.09 calories per pound per minute ≈ 405 total calories burned.

50% of those 405 calories are fat, and 50% are carbohydrate:
0.5 × 405 calories ≈ 203 each of fat and carbohydrate calories.

The runner burned many more fat calories (203 versus 125) even though a smaller percentage of calories burned were fat calories (50% versus 75%).

Formulas link to here link to index

Exercise Heart Rate

There are standard formulas for estimating a person's heart rate categories. Your maximum safe heart rate decreases with age. When doing aerobic exercise you want your heart rate between 50% and 85% of this maximum.

Heart Rate Formulas

maximum safe heart rate = 220 − age 

lower limit for aerobic exercise = maximum safe heart rate × 0.5 

upper limit for aerobic exercise = maximum safe heart rate × 0.85 

These formulas are not accurate for all people. An actual value can be measured at a doctor's office by doing different exercises while monitored by an ECG machine.

12. Arthur is 35 years old. What is his maximum safe heart rate, minimum aerobic exercise heart rate, and maximum aerobic exercise heart rate?

12. maximum safe heart rate = 220 − age = 220 − 35 = 185 beats per minute.

lower limit for aerobic exercise = 185 × 0.5 ≈ 93 beats per minute.

upper limit for aerobic exercise = 185 × 0.85 ≈ 157 beats per minute.

Basal Metabolic Rate (BMR)

sedentary sitting

In 1918, two scientists named J. Arthur Harris and Francis G. Benedict discovered that how many calories a person burns (in one day) while at rest is a feature of metabolism that can be predicted quite accurately by measuring the surface area of the person's skin. They named this resting amount of calorie use the Basal Metabolic Rate, or BMR.

The BMR measures how much energy our metablism uses if we were completely inactive and lay in bed all day.

Those two scientists then found an estimate for the BMR based on weight, height, age, and sex. This was much more useful because measuring the surface area of a person's skin is difficult. Their formulas were used for more than sixty years. But we will not use them in our math class.

The Harris-Benedict BMR Equations

Women's BMR = (weight × 4.3) + (height × 4.3) − (age × 4.7) + 655

Men's BMR = (weight × 6.2) + (height × 12.7) − (age × 6.8) + 65

These formulas use weight measured in pounds and height measured in inches. The numbers would be different if we used weight measured in kilograms and height measured in centimeters!

In 1990, these formulas were updated by Mark D. Mifflin and Sachiko T. St. Jeor. For most people these new equations are the most accurate. Use these on all class work instead of the ones above.

The Mifflin-St.Jeor BMR Equations

Women's BMR = (weight × 4.55) + (height × 15.88) − (age × 5) + 5 

Men's BMR = (weight × 4.55) + (height × 15.88) − (age × 5) − 161 

These formulas use weight measured in pounds and height measured in inches. The numbers would be different if we used weight measured in kilograms and height measured in centimeters!

Here is a list of these and other BMR formulas translated into SI units, with notes about their original research populations. For example, the The Mifflin-St.Jeor BMR Equations do tend to overestimate the BMR for young Hispanic women and for Asian women.

13. Arthur is a 35-year-old man who is moderately active, weighs 150 pounds, and is 5' 8" tall. What is his BMR?

13. Use the Mifflin-St.Jeor BMR Equation for men:
(150 lbs × 4.55) + (68 inches × 15.88) − (35 years × 5) − 161
      ≈ 1,426 calories per day

14. Odette is a 30-year-old very active woman who weighs 110 pounds and is 5' 1" tall. What is her BMR?

14. Use the Mifflin-St.Jeor BMR Equation for women:
(110 lbs × 4.55) + (61 inches × 15.88) − (30 years × 5) + 5
      ≈ 1,324 calories per day

Daily Calorie Intake (DCI)

portions of DCI

Fortunately, we are not completely lazy and do move about during the day.

The Daily Calorie Intake, or DCI, is how many calories a person really burns each day.

A person's DCI is always bigger than their BMR because they move about during the day and use more energy. So we need to multiply the BMR by something to get the DCI.

What is this scale factor? Does the way a typical person moves about during the day burn twice as many calories as the BMR? Three times the BMR?

DCI Equations

DCI = BMR × scale factor

The World Health Organization has discovered that if you ask people to rate their typical activity level only three categories are needed to get an appropriate scale factor.

DCI Scale Factors

minimal
(couch potato)
moderate
(normal)
very active
(exercise routines)
Women1.561.641.82
Men1.551.782.10

Thus an athletic woman each day burns about her BMR × 1.82 calories.

We can also think about the DCI as the number of calories a person would eat each day to maintain their current weight. If the person eats less than their DCI they lose weight. If they eat more than their DCI they gain weight.

So an athletic woman could eat BMR × 1.82 calories each day and maintain her current weight.

(For the sake of completeness, know that none of these BMR or DCI formulas work for professional atheletes. Doing that much exercise changes how metabolisms function on a deeper level. The "athletic" category is for people who exercise quite often but are not professional atheletes.)

15. What is Arthur's DCI?

15. Arthur's BMR was 1,426 calories per day. For a moderately active man the scale factor is 1.78.
BMR × 1.78 = 1,426 × 1.78 ≈ 2,539 calories per day

16. What is Odette's DCI?

16. Odette's BMR was 1,324 calories per day. For a very active woman the scale factor is 1.82.
BMR × 1.82 = 1,324 × 1.82 ≈ 2,410 calories per day

Sad News for Increased Exercise

People of all ages should be wary of hoping a new exercise program will by itself cause them to lose weight. For most people, doing additional exercise causes an increase in appetite that nearly cancels out the extra calories burned in the new exercise program. An effective weight loss program needs to change both exercise and diet.

Good News for Older People

After strenuous exercise your metabolism stays elevated for a while. This his little long-term effect on younger people. But for older people (over 50) doing resistance exercise as well as aerobic exercise noticeably raises the Basal Metabolic Rate. The BMR for older people can increase by as much as 30% if they are active.

Homework link to here link to index

Not So Fast, My Friend

Go back and actively read these online lecture notes a second time.

• Each time you get to an example problem, try to solve the problem yourself before looking at the solution.

• If you get stuck, only look at the solution one line at a time. Either scroll down very carefully or hold a piece of paper over the screen to block the lower text.

• If you are still stuck, contact the instructor or bring questions to class!

Remember the three levels of math understanding. Reviewing the notes actively can work as prompt homework problems where the goal is merely to find out how much time you will need for studying and how to pace yourself—not yet to finish problems and master topics.

More Practice

17. One 1-cup serving of 2% lowfat milk has 5 grams of fat, 12 grams of carbohydrates (all 12 from sugar), and 9 grams of protein. Change to calories the amounts of fat, carbohydrate, sugar, and protein.

17. The milk has 5 × 9 = 45 calories from fat.

Its has 12 × 4 = 48 calories from carbohydrates.

(All of those 48 calories from carbohydrates are sugar.)

It has 9 × 4 = 36 calories from protein.

(By the way, milk tastes good because it contains a lot of natural sugar!)

18. Caroline is an 80 year old woman, not physically active, who weighs 120 pounds and is 5' 3" tall. She walks for 15 minutes to the grocery store, shops for 20 minutes, and walks 15 minutes home. How many calories does she burn?

18. For Caroline's walking we multiply:
120 pounds × 30 minutes × 0.037 calories per pound per minute ≈ 133 calories.

For Caroline's shopping lifting we multiply:
120 pounds × 20 minutes × 0.028 calories per pound per minute ≈ 67 calories.

So Caroline's total is about 133 + 67 = 200 calories.

An old woman from Game of Thrones

19. What is Caroline's maximum safe heart rate, minimum aerobic exercise heart rate, and maximum aerobic exercise heart rate?

19. maximum safe heart rate = 220 − age = 220 − 80 = 140 beats per minute.

lower limit for aerobic exercise = 140 × 0.5 ≈ 70 beats per minute.

upper limit for aerobic exercise = 140 × 0.85 ≈ 119 beats per minute.

20. What is Caroline's BMR?

20. Use the Mifflin-St.Jeor BMR Equation for women:
(120 lbs × 4.55) + (63 inches × 15.88) − (80 years × 5) + 5 ≈ 1,151 calories per day

21. What is Caroline's DCI?

21. Caroline's BMR was 1,151 calories per day. For a not physically active woman the scale factor is 1.56.
BMR × 1.56 = 1,151 × 1.56 ≈ 1,796 calories per day

22. Throughout the day Caroline drinks 3 cups of 2% milk. How many calories of fat is this?

22. We just found that one cup of milk has 45 calories from fat. So three cups has 45 × 3 = 135 calories from fat.

Random Homework Problems

Homework

Each time you load the page these problems change!

Icing on the Cake

Icing on a Gingerbread House

These homework problems have no answers. First do the other homework for as much practice as you need. Once you have confidence, turn in these problems to be graded.

26. One 38-gram serving of dark chocolate has 15 grams of fat, 19 grams of carbohydrates (10 from sugar), and 3 grams of protein. Change to calories the amounts of fat, carbohydrate, sugar, and protein.

27. Zachary is a 35 year old man, moderately active, who weighs 150 pounds and is 5' 7" tall. He bikes for 15 minutes to the gym, lifts weights for 30 minutes, and then bikes 15 minutes home. How many calories does he burn?

28. What is Zachary's maximum safe heart rate, minimum aerobic exercise heart rate, and maximum aerobic exercise heart rate?

29. What is Zachary's BMR?

30. What is Zachary's DCI?