Math 20 Math 25 Math Tips davidvs.net

# Math Lecture NotesArithmetic Homework

## Small Questions

### Formulas

1. Toothpicks make squares of increasing size. What is the pattern for how many toothpicks are in each square? (We are not looking at the area of the squares.) This pattern might seem hard until you notice a trick.

1. We studied the triangle pattern as example problem six of the Arithmetic Facts page.

This pattern went 1, 3, 6, 10, ... with each step increasing additively by one more than the previous step. We found out its formula was y = x × (x + 1) ÷ 2.

We also saw that it often its the hidden core of many other real-life patterns. In this case, we are simply multipying it by four. This replaces the concluding ÷2 with a ×2. (We had half of something. Doing ×4 now gives us double that thing.)

So the new formula is y = x × (x + 1) × 2.

We can plug in 1, 2, 3, 4, ... to double-check that this does indeed give us 4, 12, 24, 30, ... as desired.

LCC Math 25 Packet Homework

Also do some arithmetic review in the packet on pages:
• PF-16 to PF-17
• MS-1 to MS-12
• MS-42 to MS-45

### Calories

#### Calories in a Gram

2. 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?

3. 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?

4. 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.

5. 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.

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

7. Brianna's daughter eats a bowl of cereal as an after-school snack: one serving of Lite Chocolate Frosted Sugar Bombs with one cup of 2% lowfat milk. How many calories of sugar is this?

2. 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.

3. 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.

4. The cereal has 1.3 × 9 ≈ 12 calories from fat. Its has 22 × 4 = 88 calories from carbohydrates. It has 9 × 4 = 36 calories 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!)

5. The milk has 5 × 9 ≈ 45 calories from fat. Its has 12 × 4 = 48 calories from carbohydrates (all sugar). It has 9 × 4 = 36 calories from protein.

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

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

7. We just found that one serving of that cereal had 36 calories from sugar, and one cup of milk had 48 calories from sugar. The total is 36 + 45 = 81 calories from sugar.

(By the way, this problem is odd from a real-life perspective. The American Heart Association recommends that children ages four to eight not have more than 130 calories of added sugar per day. But the milk's sugar is a natural part of the milk and outside of a math problem should not be counted in the same way as the cereal's added sugar.)

#### Burning Calories

8. Oscar weighs 190 pounds. He walks for 15 minutes to the grocery store, shops for 20 minutes, and walks 15 minutes home. How many calories does he burn?

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?

10. Otto weighs 155 pounds. He runs for an hour and then does a half hour of weight lifting. How many calories does he burn?

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

8. Oscar walks for a total of 30 minutes. That burns 190 pounds × 30 minutes × 0.037 calories per pound per minute ≈ 211 calories from walking. For the shopping we multiply 190 pounds × 20 minutes × 0.028 calories per pound per minute ≈ 106 calories from shopping. So the total is about 317 calories

9. Odette bicycles for a total of 30 minutes. That burns 110 pounds × 30 minutes × 0.045 calories per pound per minute ≈ 149 calories from bicycling. For the basketball we multiply 110 pounds × 120 minutes × 0.063 calories per pound per minute ≈ 832 calories from basketball. So the total is about 981 calories

10. For the running we multiply 155 pounds × 60 minutes × 0.09 calories per pound per minute ≈ 837 calories from running. For the weight lifting we multiply 155 pounds × 30 minutes × 0.039 calories per pound per minute ≈ 181 calories from weight lifting. So the total is about 1,018 calories.

11. 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

12. Two people weigh 150 pounds. They exercise for 30 minutes: one person walks and the other person runs. How many fat and carbohydrate calories do each burn?

12. First find the total calories burned. The walker burns 150 pounds × 30 minutes × 0.037 calories per pound per minute = 166.5 calories. The runner burns 150 pounds × 30 minutes × 0.09 calories per pound per minute = 405 calories.

Next find the fat calories burned. The walker burns 75% of the total, so 166.5 calories × 0.75 ≈ 125 fat calories. The runner burns 50% of the total, so 405 calories × 0.5 ≈ 203 fat calories.

Lastly find the carobhydrate calories burned. The walker burns 25% of the total, so 166.5 calories × 0.25 ≈ 42 carobhydrate calories. The runner burns 50% of the total, so 405 calories × 0.5 ≈ 203 carobhydrate calories.

#### Basal Meabolic Rate

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

14. Brianna is a 23-year-old very active woman who weighs 130 pounds and is 5' 1" tall. What is her BMR?

15. Caroline is an 80 year old woman, not physically active, who weighs 120 pounds and is 5' 3" tall. What is her BMR?

13. Use the formula for a man's BMR to find (150 lbs × 4.55) + (68 inches × 15.88) − (35 years × 5) − 161 ≈ 1,426 calories per day

14. Use the formula for a woman's BMR to find (130 lbs × 4.55) + (61 inches × 15.88) − (23 years × 5) + 5 ≈ 1,450 calories per day

15 Use the formula for a woman's BMR to find (120 lbs × 4.55) + (63 inches × 15.88) − (80 years × 5) + 5 ≈ 1,151 calories per day

#### Daily Calorie Intake

16. What is Arthur's DCI?

17. What is Brianna's DCI?

18. What is Caroline's DCI?

16. For a moderately active man the scale factor is 1.78, so BMR × 1.78 = 1,426 × 1.78 ≈ 2,539 calories per day

17. For a very active woman the scale factor is 1.56, so BMR × 1.56 = 1,450 × 1.82 ≈ 2,639 calories per day

18 For a not physically active woman the scale factor is 1.56, so BMR × 1.56 = 1,151 × 1.56 ≈ 1,796 calories per day

### Body Fat

#### Body Mass Index

19. What is Arthur's BMI?

20. What is Brianna's BMI?

21. What is Caroline's BMI?

19. BMI = weight ÷ height2 × 703 = BMI = 150 ÷ 682 × 703 ≈ 22.9

20. BMI = weight ÷ height2 × 703 = BMI = 130 ÷ 612 × 703 ≈ 24.6

21. BMI = weight ÷ height2 × 703 = BMI = 120 ÷ 632 × 703 ≈ 21.3

(By the way, notice that BMI numbers lack unit labels. That is clue that it does not measure anything reliable in real life. Also notice that our only very active person had the highest BMI!)

#### Percent Body Fat

22. What is Arthur's estimated percent body fat?

23. What is Brianna's estimated percent body fat?

24. What is Caroline's estimated percent body fat?

19. For men, percent body fat is roughly (1.2 × BMI) + (0.23 × age) + 5.4 = (1.2 × 22.9) + (0.23 × 35) + 5.4 ≈ 40.8

20. For women, percent body fat is roughly (1.2 × BMI) + (0.23 × age) − 5.4 = (1.2 × 24.6) + (0.23 × 23) − 5.4 ≈ 29.4

(By the way, these numbers classify Arthur and Caroline as "obese" even though nothing about their height, weight, and amount of physical activity would lead us to believe they are not average.)

21. For women, percent body fat is roughly (1.2 × BMI) + (0.23 × age) − 5.4 = (1.2 × 21.3) + (0.23 × 80) − 5.4 ≈ 38.6

(By the way, these numbers classify Brianna as "average". This is probably just as innacurate. But it could be reasonable, considering her height and weight, if her physical activity gives her strong thighs rather than unusually well-defined abdominal muscles. Perhaps her above-average activity level is from commuting by bicycle?)

### Exercise Pulse Rate

25. What is Arthur's maximum safe heart rate, minimum aerobic exercise heart rate, and maximum aerobic exercise heart rate?

26. What is Brianna's maximum safe heart rate, minimum aerobic exercise heart rate, and maximum aerobic exercise heart rate?

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

25. For Arthur:
maximum safe heart rate = 220 − age = 220 − 35 = 185 beats per minute
upper limit for aerobic exercise = maximum safe heart rate × 0.85 ≈ 157 beats per minute
lower limit for aerobic exercise = maximum safe heart rate × 0.5 ≈ 93 beats per minute

26. For Brianna:
maximum safe heart rate = 220 − age = 220 − 23 = 197 beats per minute
upper limit for aerobic exercise = maximum safe heart rate × 0.85 ≈ 167 beats per minute
lower limit for aerobic exercise = maximum safe heart rate × 0.5 ≈ 99 beats per minute

27. For Caroline:
maximum safe heart rate = 220 − age = 220 − 80 = 140 beats per minute
upper limit for aerobic exercise = maximum safe heart rate × 0.85 ≈ 119 beats per minute
lower limit for aerobic exercise = maximum safe heart rate × 0.5 ≈ 70 beats per minute

### Three Kinds of Averages

28. The home values on a certain street, in thousands of dollars, are: 384, 364, 342, 346, 360, 356, 265, 417, and 530. What is the mean, median, and mode of these numbers? Which type of average best communicates the "typical" home on that street?

29. A shipping company needs to transport seven freight containers. Their weights are 10, 16, 16, 18, 20, 60, and 77 tons. What is the mean, median, and mode of these numbers? Which type of average best communicates the "typical" weight of the containers?

30. Two company clerks receive a report that only contains the mean and median weights, and number of containers, from the previous problem. The first clerk tries to find the total weight by multiplying the mean by the number of containers. The second clerk tries to find the total weight by multiplying the median by the number of containers. Which clerk is correct? Why? How much error does the other clerk have?

28. For the home values:

The sum is \$3,364,000. There are nine homes. Divide to find the mean, which is \$374,000.

The sorted list is 265, 342, 346, 356, 360, 364, 384, 417, and 530. So the median is the middle value of 360, corresponding to \$360,000.

No value repeats, so all the values are modes!

(By the way, the median is more traditional for home values so one atypical neighbor does not affect the local values as much. But someone selling a house on the street might use the mean so the neighboring property values sound higher.)

29. For the shipping containers:

The sum is 217 tons. There are seven containers. Divide to find the mean, which is 31 tons.

The list is already sorted, and the middle value is 18 tons. In this particular case this is most "typical" for five of the seven shipping containers.

Only the value 16 repeats, so the mode is 16 tons.

30. The first clerk gets 31 × 7 = 217 tons, which is correct. He "worked backwards" from the mean to the total. The second clerk gets 18 × 7 = 126 tons, which is too light by 217 − 126 = 91 tons. The median neglected the two much heavier containers which made his estimate is too small.

## Big Questions

The "big questions" in each homework assignment represent real-life situations. Sometimes they require researching additional facts not included in the problem or the Math 25 website.

B1. Donna is a 19-year-old moderately active woman who weighs 120 pounds and is 5' 4" tall. What is her BMR, DCI, and percent body fat? How should Donna make use of the USDA recommendations for diets with 2,000 and 2,500 calories per day? What is her maximum safe heart rate, minimum aerobic exercise heart rate, and maximum aerobic exercise heart rate? If she walks for an hour and then rides her bike for half an hour, how many calories does she burn—and to how many York Peppermint Pattie candies is that many calories equivalent?

B2. What is your own BMR, DCI, and percent body fat? How should your make use of the USDA recommendations for diets with 2,000 and 2,500 calories per day? What is your maximum safe heart rate, minimum aerobic exercise heart rate, and maximum aerobic exercise heart rate? Pick one type of exercise you do and find how many calories you burn doing it.

B3.The following recipe is adapted from the website Chocolate and Zucchini. Convert this recipe from a restaurant-size recipe to a home kitchen recipe. Scale down the recipe so it makes only six servings, and uses cups of skim milk and cups of sugar. Your smaller recipe will still use a single vanilla bean, rather than a fractional amount.

B4. During the 2007 strike of the Writer's Guild of America, two different news reports painted very different pictures of these screen and television writers.

• According to CNBC, there were 4,434 guild writers who worked full-time in 2006, and their average salary was \$204,000. (CNBC headline, October 11, 2007)

• According to the Los Angeles Times, the median income of the writers from their guild-covered employment is \$5,000 a year. (Howard A. Rodman, October 17, 2007)

Which was true? Were Hollywood's writers very wealthy and going to strike even though they earned much more than most Americans? Or were they poor and going on strike to defend the few thousand dollars they could earn from their writing? The truth was that a very few writers earn millions of dollars (what did that do to the mean?) and almost half of the guild's writers don't write anything in a given year (what does having half the list values zero do to the median?) How could averages been used better to explain and report that situation?

B5. All the rooms in the floor plan below have ceilings 7' 6" high. The bedrooms and living room are carpeted. The bathroom, kitchen, and dining room have vinyl floors. (a) At \$3 per square foot, what is the cost of replacing all the vinyl flooring? (b) Which of the three carpet cleaning services is the best buy, and how much will it cost? (c) If you need to paint the three non-closet walls of the first bedroom, what is the area that you will paint? (The balcony has a plastic "fake grass" surface. The closets are carpeted, but do not need cleaning. The T-shaped hallways has tile. Those areas are not part of our problem.)

B6. Gina's mother keeps pestering her to "shop for a husband while you are young and pretty." Gina's friends assure her that this advice is outdated, and Gina can focus on her career at least until the age of 30 if not 35. Use the graph of census data to decide who is correct.

B7. Mel Bartholemew is making another of his Square Foot Gardens. This garden will have a 20 square foot tomato plant section (only tomato plants), a 16 square foot "soup greens" section (half Swiss chard and half kale), and a 24 square feet "root vegetables" section (half carrots and half onions). He always plants one tomato plant per square foot (large), four Swiss chard or kale plants per square foot (medium), and sixteen carrot or onion plants per square foot (small). Weather forces him to use tomato starts instead of seeds. Find the total size of his garden. Estimate the cost of his garden after contacting an appropriate store for prices.

LCC Math 25 Packet Homework

Also do some area and volume problems in the packet on pages:
• MS-17 to MS-25
• MS-29 to MS-30