Algebra
Chapter 2 Answers
Patterns, Change, and Expressions
Changes in one quantity can produce changes in related quantities. In the following activities, you'll investigate how changes in a quantity can affect the density of matter, the characteristics of human populations, and the physical properties of water.
Part A, Quantities and Change
1. Density is a measure of the weight of a substance per unit of volume.
The density of sea water is 64 pounds per cubic foot. This means that the
sea water in an aquarium with a volume of 1 cubic foot weighs 64 pounds.
a. Click on the element manganese (Mn) in the Periodic table of the elements Web site. The number 25 represents the number of protons in the nucleus of an atom of manganese. Choose "Bulk properties" in the list Manganese: the element. What is the density of manganese? Hint: kg m3 means "kilograms per cubic meter."
[7470 kg/m3]b. Is the density of manganese, as given in the table, constant or variable?
[The density of manganese is constant. Students may be interested to know that density in fact varies slightly with temperature, distance from the center of the earth, and other factors. Densities in the table, however, are given for standardized conditions, which provide for constant values.]c. Find the densities of iron (Fe, atomic number 26) and cobalt (Co, atomic number 27).
[iron, 7874 kg/m3; cobalt, 8900 kg/m3]d. Is the property of matter known as density variable or constant? Explain.
[Variable. Results in c show that different materials may have different densities. Remind students that, while density itself varies, the density of a given material, under standardized conditions, is constant (see b above).]e. The cubes are made of iron. What is the weight of each?
[62,992 kg; 212,598 kg]f. Is weight a constant or a variable? Explain.
[Variable. The weight of an object depends on many factors, including its size and its density.]g. Name a property of the cube on the left, above, that is constant. Then name a property that may vary. (Choose properties other than weight or density.)
[Possible answers include: The dimensions are constant. The temperature may vary.]
Part B, Describing Change
2. Massive amounts of data on the geography, people, governments, and economies of the world have been collected for the World Factbook Web site.
a. Click on "Countries" and complete the table.
Argentina
Belgium
Malawi
Vietnam
PEOPLE
1
Population growth rate (%)1.3
0.1
1.57
1.51
2
Life expectancy at birth (total population)74.31
77.19
35.26
67.38
3
Literacy: total population (%)96.2
99
56.4
93.7
ECONOMY
4
GDP: per capita ($/yr)8600
20,300 800
1470
5
GDP: composition by sector (agriculture)
(%)7
2
45
28
6
Electricity: consumption per capita (kWh)1606
6823
67
154
b. Are the quantities in Column 1 and Column 2 directly or inversely related? How can you tell?
[Inversely. As growth rate decreases, life expectancy increases.]c. What reasons might explain the relationship you observed in b?
[Answers will vary. A decrease in birth rate is likely to free up more resources for the population, leading to better health and longer life for citizens.]d. GDP stands for "gross domestic product," a measure of a country's annual economic growth. Are the quantities in Column 4 and Column 6 directly or inversely related? How can you tell?
[Directly. As per capita GDP increases, per capita electric consumption increases too.]e. What reasons might explain the relationship you observed in d?
[Answers will vary. As a person's share of the country's wealth increases, he or she is likely to choose to spend more money on goods and services provided by electricity and electrical products.]f. Column 5 gives the portion of a country's growth earned through agriculture. Are the quantities in Column 3 and Column 5 directly or inversely related?
[Inversely]g. What reasons might explain the relationship you observed in f?
[Answers will vary. Education is often neglected in agricultural nations because farm laborers may not need to be highly educated. As countries become more industrialized, however, their governments may need to provide citizens with more education--and higher rates of literacy--so that they can take on more sophisticated jobs.]
Part C, Graphing Change
3. Find the table Boiling Point of Water vs. Altitude on the Web.
a. If you traveled gradually from sea level (0 ft altitude) to 10,000 ft altitude, how would air pressure change during your journey? How would the boiling point of water change?
[Both would decrease.]b. State whether the following quantities are directly or inversely related:
- · altitude and air pressure
· altitude and boiling point of water
· air pressure and boiling point of water- [inversely; inversely; directly]
c. Graph altitude versus boiling point of water. Place altitude (ft) on the horizontal axis and boiling point (° F) on the vertical axis. Use altitude values from 0 ft to 10,000 ft in intervals of 1,000 ft.
d. Contaminated water can be purified (sterilized) by boiling. To see how sterilization time changes with altitude, see Paragraph 4 under Dehydration at the Military Field OperationsWeb site. Graph altitude versus sterilization time. Place altitude (ft) on the horizontal axis and sterilization time on the vertical axis. Use altitude values from 0 ft to 20,000 ft in intervals of 2,000 ft.
e. Are altitude and water sterilization time directly or inversely related? [Directly]
Part D, Making Connections
4. Refer again to the table Boiling Point of Water vs. Altitude to answer the following questions.
a. When you examine a table of values, how can you determine whether two quantities are directly or inversely related?
[If the values of both quantities increase or decrease together, the quantities are directly related. If one increases as the other decreases, the quantities are inversely related.]b. Look again at the graphs you drew for 3c and 3d. When you examine a graph, how can you determine whether two quantities are directly or inversely related?
[A graph that slants from the upper left to the lower right indicates that as one quantity decreases the other increases, which is an inverse relationship. A graph that slants from the lower left to the upper right indicates that as one quantity increases the other also increases, which is a direct relationship.]
Relationships in science are often expressed using the language of algebra.
In the following explorations, you'll use algebra to investigate temperature,
an important topic in the science of physics.
Part A, Using Variables and Expressions
1. You're probably familiar with the Fahrenheit and Celsius temperature scales. Temperatures in science are often expressed using the Kelvin scale. Read the first paragraph on the Scales of temperature Web site then answer the questions below.
a. Using the rounded figure instead of the "precise" figure, describe the relationship between the Celsius and Kelvin scales.
[The Kelvin temperature is the Celsius temperature plus 273 degrees.]b. If the air temperature is C degrees Celsius, what is the Kelvin temperature? [C + 273]
c. If the temperature of a liquid is n degrees Kelvin, what is the Celsius temperature? [n - 273]
d. Complete the table.
Celsius (C)
0
10
20
30
40
50
Kelvin (K)
273
283
293
303
313
323
e. Write each pair of values as an ordered pair. Then plot the ordered pairs on a coordinate plane.
[(0,273); (10,283); (20,293); (30,303); (40,313); (50,323)]
f. Are Celsius and Kelvin temperatures directly or inversely related? How do you know?
[Directly. As one increases, the other increases.]
Part B, Evaluating Expressions
2 . One formula relating Celsius and Fahrenheit temperatures is
C = f(5(F 32),9) where C is Celsius temperature and F is Fahrenheit
temperature. Find the hottest temperature ever recorded on earth on the
Death Valley
Web site.
a. What is the hottest temperature in degrees Celsius? Round to the nearest degree. [136°F, 58°C]
b. Write the earth's record hot temperature in degrees Kelvin.
[331°K]c. Celsius and Fahrenheit temperatures are also related by the formula F = 1.8 C + 32. Find the boiling point of copper at the Web elements site. Write the temperature in degrees Celsius. Then convert it to the Fahrenheit scale. [297°C, 5300.6°F]
d. Sound travels faster in hot air than it does in cool air. The velocity of sound and Kelvin temperature are related by the formula K = f(v2,404), where K is Kelvin temperature and v is the speed of sound in meters per second. Find the Kelvin temperature at the surface of the planet Mercury, where on a hot day sound may travel 560 meters per second. [784°K]
e. Write Mercury's surface temperature in degrees Celsius and degrees Fahrenheit. [511°C, 951.8°F]
Use the grammar of algebra to look at some formulas related to light trucks.
Part A, Order of Operations
1. Find out how much extra ground clearance will be provided under a light truck's axles if you change the size of your tires. Look at the site for Light Truck Fitment Formulas.
a. Suppose the diameter of your new tire is 33" and the diameter of your old tire is 28". Calculate the approximate change in ground clearance and center of gravity.
[(33" - 28")/2 = 2.5" increase in ground clearance and center of gravity.]
b. Suppose you saw your friend calculate the answer to a like this: 33 - (28/2 ). Explain what is wrong with this calculation and why.
[The parentheses in the expression in a indicate that we are to subtract the two diameters, take that result and then divide by 2. The student in this problem only divided the original tire diameter by 2.]
c. When you change the tire size, it affects the speedometer. Calculate how far off the speedometer will be if you replace your original tires, which have a 28" diameter, with new tires, which have a 33" diameter, when the indicated speed is 65 mph. [(33"/28") x 65 mph = 76.6 mph.]
Part B, The Distributive Property
2. Look again at the section entitled "How much extra ground clearance under the axles will be provided?"
a. Explain why 32.5"/2 - 27"/2 gives the same result as (32.5" - 27")/2.
[This is an example of the distributive property. It shows division over subtraction.]
b. Use the distributive property to explicitly rewrite the problem in a to show why the two expressions are equal.
[(32.5 - 27)/2 = 1/2 · (32.5 - 27) = 1/2 · 32.5 - 1/2 · 27 = 32.5/2 - 27/2]
3. Sometimes, if you need new tires, you also need new wheels.
a. If tires are $60 each and wheels are $120 each, write two different expressions that show the total cost of getting 4 tires and 4 wheels. [4 · 60 + 4 · 120, 4 · (60 + 120)]
b. Calculate the answers to the expressions you wrote in a. [$720]
c. Are both your answers from b the same? Explain.
[Yes. The expressions are just two different ways of writing the same thing.]
d. Which expression was easier to calculate? Explain. [Answers may vary.]
Part C, Simplifying Expressions
4. Look at the expression for the approximate change in ground clearance and center of gravity: (New Tire Diameter - Original Tire Diameter) /2
a. What are the terms in the expression?
[The terms are the New Tire Diameter and the Original Tire Diameter. Terms are the parts of the expression that are added. Remember subtraction is the same as adding the opposite signed number.]
b. Is there a constant in the expression? If so, what is it?
[The constant in the expression is multiplying by 1/2 or dividing by 2.]
5. Suppose you are going to change the oil in your car. To do this you need 8 quarts of oil and 1 oil filter.
a. If n is the cost of a quart of oil and f is the cost of an oil filter, write an expression that shows the total amount you will spend buying oil and a filter. [8n + f]
b. What is the coefficient of n? [8]
c. What does this represent?
[The number of quarts of oil purchased]
d. What is the coefficient of f? [1 ]
e. What does this represent?
[The number of oil filters purchased]
Part D, Making Connections
6. Look at the section entitled "How far off
will the speedometer be?" As the new tire diameter increases, what
happens to the actual speed? Make a chart showing at least 3 calculations
to justify your conjecture.
[As the new tire diameter increases, so
does the actual speed.]
New Tire Diameter |
Old Tire |
Indicated |
Actual |
32.5 in. |
27 in. |
55 mph |
66.2 mph |
33 in. |
27 in. |
55 mph |
67.2 mph |
33.5 in. |
27 in. |
55 mph |
68.2 mph |
7. Let w = indicated speed, x = new tire diameter, y = original tire diameter, and z = original axle ratio.
a. Write an expression in terms of w, x, y, and z for how far off the speedometer will be. [(x/y) w = actual speed]
b. Write an expression for the required axle ratio using the information found in the section entitled "What can I do to regain the performance?" [(x/y) z = required axle ratio]
c. Can you simplify your expressions in a and b? Explain.
[The expressions do not have like terms and cannot be simplified. However, they can be multiplied and shown in a different form: xw/y and xz/y.]