Lesson 2.5
Introduction
Course Objectives
This lesson will address the following course outcomes:
· 10. Apply quantitative reasoning strategies to solve real-world problems with proportional relationships using whole numbers, fractions, decimals, and percents as appropriate.
Specific Objectives
Students will understand
· the concept of population density as a ratio.
· what is meant by proportional or change based on a constant ratio.
Students will be able to
· estimate between which two powers of 10 a quotient of large numbers lies.
· calculate a unit rate.
· solve a proportion by first finding a unit rate and then multiplying appropriately.
Estimating Densities
Problem Situation: Estimating Population Densities
In this lesson, you will compare the populations of different states.
#1 Points possible: 20. Total attempts: 5
Use the information given below to calculate the population density for each of the states listed, in people per square mile. Round to the nearest tenth.
State | Population | Land Area Square Miles | Population density people/mi2 |
Alaska | 710,231 | 571,951 | |
Idaho | 1,567,582 | 82,747 | |
Kentucky | 4,339,367 | 39,728 | |
Louisiana | 4,533,372 | 43,562 | |
Nebraska | 1,826,341 | 76,872 | |
New Hampshire | 1,316,470 | 8,968 | |
New Mexico | 2,059,179 | 121,356 | |
South Dakota | 814,180 | 75,885 | |
Washington | 6,724,540 | 66,544 | |
Wisconsin | 5,686,986 | 54,310 |
#2 Points possible: 20. Total attempts: 5
The states can be divided up into groups based on their densities. Determine which of the four given groups each of the states belongs in.
State | Density Group | |
Alaska | ||
Idaho | ||
Kentucky | ||
Louisiana | ||
Nebraska | ||
New Hampshire | ||
New Mexico | ||
South Dakota | ||
Washington | ||
Wisconsin |
#3 Points possible: 30. Total attempts: 5
Think about strategies you can use to estimate the population density of the states without using a calculator. Use your strategies to divide these states into one of the four density groups.
State | Population | Land Area Square Miles | Density Group | |
Georgia | 9,687,653 | 57,906 | ||
Kansas | 2,853,118 | 81,815 | ||
Montana | 989,415 | 145,552 | ||
Nevada | 2,700,551 | 109,826 | ||
New Jersey | 8,791,894 | 7,417 | ||
Oregon | 3,831,074 | 95,997 | ||
Rhode Island | 1,052,567 | 1,045 | ||
South Carolina | 4,625,364 | 30,109 | ||
Tennessee | 6,346,105 | 41,217 | ||
Wyoming | 563,626 | 97,100 |
#4 Points possible: 10. Total attempts: 5
Find which state from the previous two problems has the greatest population density. Hint
State:
Calculate that population density. Round to the nearest tenth.
Population Density: people/square mile
#5 Points possible: 10. Total attempts: 5
Find which state from the previous problems has the least population density.
State:
Calculate that population density. Round to the nearest tenth.
Population Density: people/square mile
Scaling Densities
If we knew population density and land area, we could use this to estimate the population. We’ll explore that in the next few problems.
#6 Points possible: 10. Total attempts: 5
Tacoma, WA has a population of about 200,000 and covers approximately 62 square miles. New York City has a population of about 8.4 million, and covers approximately 469 square miles. If Tacoma had the same population density as New York City, what would the population of Tacoma be (accurate to the nearest hundred thousand)?
Try the problem on your own first. If you are having trouble after 2 tries, we will break it down.
The population of Tacoma would be people
#7 Points possible: 5. Total attempts: 5
Most of the world outside the United States uses the metric system of measurement, so it is often useful to be able to make comparisons between the American system and the metric system.
Bangladesh has a population density of 1,127 people/square kilometer. 1 kilometer = 0.62 mile.
Suppose you converted the density of Bangladesh to square miles. Would a square mile contain more people, less people, or the same number of people as a square kilometer?
· More people
· The same number of people
· Less people
#8 Points possible: 5. Total attempts: 5
Which of the following statements is the most accurate description of the relationship between a square kilometer and a square mile?
· A square kilometer is about one-sixth of a square mile.
· A square kilometer is about two-thirds of a square mile.
· A square kilometer is about one-third of a square mile.
· A square kilometer is about six-tenths of a square mile.
#9 Points possible: 10. Total attempts: 5
Now let’s combine the ideas from the last three questions. Tacoma, WA has a population of about 200,000 and covers approximately 62 square miles. Bangladesh has a population density of 1,127 people/square kilometer. How many people would be in Tacoma if the population density were the same as Bangladesh? Give an answer accurate to the nearest hundred people.
Try the problem on your own first. If you are having trouble after 2 tries, we will break it down.
The population of Tacoma would be people
HW 2.5
#1 Points possible: 5. Total attempts: 5
Which of the following was one of the main mathematical ideas of the lesson?
· You can find a new value that is changing by a constant rate by multiplying by a ratio. Example: A person’s wage is $10.35/hour. How much does the person earn in 40 hours? $10.35/hour × 40 hours = $414.
· A percent is a ratio compared to 1. Example: 12% increase in population means that the number of people increases by 12 for every 1 person in the original population.
· Population density measures how crowded a state is. Example: 2.3 people per square mile is less dense than 8.7 people per square mile.
· You can find a new value that is changing by a constant rate by adding a ratio. Example: A car’s gas mileage is 20 miles/gallon. How far can it drive on 5 gallons of gasoline? 20 mi/gal + 5 = 25 miles.
#2 Points possible: 5. Total attempts: 5
In the lesson, you used ratios in the form of population densities. A population density is a ratio because it is a comparison of two measures: Number of people per number of square miles. Which of the following are ratios? There may be more than one correct answer.
· $98
· 5 lb/$3
· 67 hours
· 10 miles/hour
· 252 miles
#3 Points possible: 5. Total attempts: 5
Calculate the gas mileage of a car that drives 283 miles on 12.3 gallons of gas. Round to the nearest tenth of a mile/gallon. miles per gallon
#4 Points possible: 5. Total attempts: 5
A car drives 630 miles on 35 gallons of gas. How far can it drive on 12 gallons? miles
#5 Points possible: 5. Total attempts: 5
A jar holds 128 fluid ounces of juice. The label says the jar has 16 servings. How many fluid ounces are needed for 80 servings?
fluid ounces
#6 Points possible: 15. Total attempts: 5
According to the oil company BP, in 2010, the United States used 19,148,000 barrels of oil a day, and worldwide around 87,382,000 barrels of oil per day were used.1 This includes oil used for (among other things) fuel and manufacturing.
a. If there were 309 million people in the United States in 2010, what was the daily consumption rate per person in the United States? Round to the nearest hundredth of a barrel. barrels per person
b. If there were 6.89 billion people in the world in 2010, which of the following statements would be correct? There may be more than one correct answer.
· If the world used oil at the same rate as the United States, it would have used about 93,450,000 barrels of oils per day.
· The U.S. rate of oil consumption per person was about five times the world rate.
· About one-fifth of the oil used in the entire world in one day is used in the United States.
· About half the people in the world lived in the United States.
c. There are 42 gallons in a barrel of oil. Which of the following statements is true? There may be more than one correct answer.
· If the world used oil at the same rate as the United States, it would use about 426,957,000 gallons of oil per day.
· The American rate of oil consumption is 7,452,380 gallons of oil per day.
· The American rate of oil consumption per person is about 2.5 gallons of oil use per day.
· The American rate of oil consumption is 5 more gallons of oil per day than the world rate.
#7 Points possible: 20. Total attempts: 5
Approximately 6.9 billion people now inhabit the earth. The surface area of the earth is 510,065,600 km2.
a. What is the surface area of the earth in mi2? Round to the nearest million square miles. Hint: If 1 km = 0.6214 mi, then 1 km2 = how many mi2?
· 317,000,000 mi2
· 1,321,000,000 mi2
· 197,000,000 mi2
· 820,000,000 mi2
b. The surface area above includes both land and water. Approximately 139 million square miles of the earth’s surface area is water. Using your answer from Part (a), determine what percentage of the surface area is land. Round to the nearest tenth of a percent. %
c. Approximately 1/3 of the land is uninhabitable, meaning people cannot live on it. How much land on the earth is habitable (can be lived on)? Round to the nearest million square miles.
· 19,000,000 mi2
· 130,000,000 mi2
· 39,000,000 mi2
· 92,000,000 mi2
d. Calculate the population density of the earth in people per square mile of habitable land. Round to the nearest person per square mile. people per square mile
#8 Points possible: 10. Total attempts: 5
Chickens that are not caged 24-hours a day are sometimes called “free-range” chickens. The United States Department of Agriculture (USDA) allows chickens to be called free-range as long as the chickens spend some of their time outside. The European Union (EU), however, has several additional restrictions. One of these is that the farmers must provide enough outside area so that if all the chickens were outside, the density of chickens would be no more than 0.25 chickens/sq meters.1
a. How many square meters does the EU require for one chicken? square meters
b. A farmer in the United States wants to meet the EU guidelines. She measures her area in yards. Rounded to the nearest square yard, how many square yards does she need for 1,100 chickens? (1 m = 1.0936 yd)
· 4,076 square yards
· 4,400 square yards
· 4,812 square yards
· 5,262 square yards
· None of the above
#9 Points possible: 5. Total attempts: 5
People often confuse the words million, billion, and trillion when speaking. An estimate can help you decide if the speaker uses the correct word. Consider this situation: A speaker says, “The U.S. federal debt is $14 billion dollars. That’s over $45,000 for every person in the country.”
Select the correct statement from the choices below. Note: When you say the numbers are consistent, you mean that they make sense in relationship to each other.
· The two numbers in the statement are not consistent. If the debt is $45,000 per person, the total debt must be $14 million.
· The two numbers in the statement are not consistent. If the debt is $45,000 per person, the total debt must be $14 trillion.
· The two numbers in the statement are consistent with each other.
#10 Points possible: 12. Total attempts: 5
Refer to Lesson 1.4 in which you compared the water footprint of different countries. The following information was given.
Country | Population
(in thousands) |
Total Water Footprint 1
(in 109 cubic meters per year) |
China | 1,257,521 | 883.39 |
India | 1,007,369 | 987.38 |
a. What does 109cubic meters mean?
· one billion cubic meters
· one hundred thousand cubic meters
· one trillion cubic meters
· one million cubic meters
b. The following equation is based on information from the table. 883.391,257,521=x1,007,369883.391,257,521=x1,007,369 What does x represent?
· x represents the population of China if it used water at the same rate as India.
· x represents the water footprint of China if it used water at the same rate as India.
· x represents the population of India if it used water at the same rate as China.
· x represents the water footprint of India if it used water at the same rate as China.
c. Solve the equation from part b. Give your answer to 2 decimal places. billion cubic meters per year