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Contemporary Mathematics

9.5 Measuring Temperature

Contemporary Mathematics9.5 Measuring Temperature
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Table of contents
  1. Preface
  2. 1 Sets
    1. Introduction
    2. 1.1 Basic Set Concepts
    3. 1.2 Subsets
    4. 1.3 Understanding Venn Diagrams
    5. 1.4 Set Operations with Two Sets
    6. 1.5 Set Operations with Three Sets
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  3. 2 Logic
    1. Introduction
    2. 2.1 Statements and Quantifiers
    3. 2.2 Compound Statements
    4. 2.3 Constructing Truth Tables
    5. 2.4 Truth Tables for the Conditional and Biconditional
    6. 2.5 Equivalent Statements
    7. 2.6 De Morgan’s Laws
    8. 2.7 Logical Arguments
    9. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  4. 3 Real Number Systems and Number Theory
    1. Introduction
    2. 3.1 Prime and Composite Numbers
    3. 3.2 The Integers
    4. 3.3 Order of Operations
    5. 3.4 Rational Numbers
    6. 3.5 Irrational Numbers
    7. 3.6 Real Numbers
    8. 3.7 Clock Arithmetic
    9. 3.8 Exponents
    10. 3.9 Scientific Notation
    11. 3.10 Arithmetic Sequences
    12. 3.11 Geometric Sequences
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  5. 4 Number Representation and Calculation
    1. Introduction
    2. 4.1 Hindu-Arabic Positional System
    3. 4.2 Early Numeration Systems
    4. 4.3 Converting with Base Systems
    5. 4.4 Addition and Subtraction in Base Systems
    6. 4.5 Multiplication and Division in Base Systems
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  6. 5 Algebra
    1. Introduction
    2. 5.1 Algebraic Expressions
    3. 5.2 Linear Equations in One Variable with Applications
    4. 5.3 Linear Inequalities in One Variable with Applications
    5. 5.4 Ratios and Proportions
    6. 5.5 Graphing Linear Equations and Inequalities
    7. 5.6 Quadratic Equations with Two Variables with Applications
    8. 5.7 Functions
    9. 5.8 Graphing Functions
    10. 5.9 Systems of Linear Equations in Two Variables
    11. 5.10 Systems of Linear Inequalities in Two Variables
    12. 5.11 Linear Programming
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  7. 6 Money Management
    1. Introduction
    2. 6.1 Understanding Percent
    3. 6.2 Discounts, Markups, and Sales Tax
    4. 6.3 Simple Interest
    5. 6.4 Compound Interest
    6. 6.5 Making a Personal Budget
    7. 6.6 Methods of Savings
    8. 6.7 Investments
    9. 6.8 The Basics of Loans
    10. 6.9 Understanding Student Loans
    11. 6.10 Credit Cards
    12. 6.11 Buying or Leasing a Car
    13. 6.12 Renting and Homeownership
    14. 6.13 Income Tax
    15. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  8. 7 Probability
    1. Introduction
    2. 7.1 The Multiplication Rule for Counting
    3. 7.2 Permutations
    4. 7.3 Combinations
    5. 7.4 Tree Diagrams, Tables, and Outcomes
    6. 7.5 Basic Concepts of Probability
    7. 7.6 Probability with Permutations and Combinations
    8. 7.7 What Are the Odds?
    9. 7.8 The Addition Rule for Probability
    10. 7.9 Conditional Probability and the Multiplication Rule
    11. 7.10 The Binomial Distribution
    12. 7.11 Expected Value
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  9. 8 Statistics
    1. Introduction
    2. 8.1 Gathering and Organizing Data
    3. 8.2 Visualizing Data
    4. 8.3 Mean, Median and Mode
    5. 8.4 Range and Standard Deviation
    6. 8.5 Percentiles
    7. 8.6 The Normal Distribution
    8. 8.7 Applications of the Normal Distribution
    9. 8.8 Scatter Plots, Correlation, and Regression Lines
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  10. 9 Metric Measurement
    1. Introduction
    2. 9.1 The Metric System
    3. 9.2 Measuring Area
    4. 9.3 Measuring Volume
    5. 9.4 Measuring Weight
    6. 9.5 Measuring Temperature
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  11. 10 Geometry
    1. Introduction
    2. 10.1 Points, Lines, and Planes
    3. 10.2 Angles
    4. 10.3 Triangles
    5. 10.4 Polygons, Perimeter, and Circumference
    6. 10.5 Tessellations
    7. 10.6 Area
    8. 10.7 Volume and Surface Area
    9. 10.8 Right Triangle Trigonometry
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  12. 11 Voting and Apportionment
    1. Introduction
    2. 11.1 Voting Methods
    3. 11.2 Fairness in Voting Methods
    4. 11.3 Standard Divisors, Standard Quotas, and the Apportionment Problem
    5. 11.4 Apportionment Methods
    6. 11.5 Fairness in Apportionment Methods
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  13. 12 Graph Theory
    1. Introduction
    2. 12.1 Graph Basics
    3. 12.2 Graph Structures
    4. 12.3 Comparing Graphs
    5. 12.4 Navigating Graphs
    6. 12.5 Euler Circuits
    7. 12.6 Euler Trails
    8. 12.7 Hamilton Cycles
    9. 12.8 Hamilton Paths
    10. 12.9 Traveling Salesperson Problem
    11. 12.10 Trees
    12. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  14. 13 Math and...
    1. Introduction
    2. 13.1 Math and Art
    3. 13.2 Math and the Environment
    4. 13.3 Math and Medicine
    5. 13.4 Math and Music
    6. 13.5 Math and Sports
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  15. A | Co-Req Appendix: Integer Powers of 10
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
  17. Index
A close-up view of a thermometer shows 19 degrees Celsius and 66 degrees Fahrenheit. The thermometer is positioned near a plant.
Figure 9.15 A thermometer that measures temperature in both customary and metric units. (credit: “Thermometer” by Jeff Djevdet/Flickr, CC BY 2.0)

Learning Objectives

After completing this section, you should be able to:

  1. Convert between Fahrenheit and Celsius.
  2. Identify reasonable values for temperature applications.
  3. Solve application problems involving temperature.

When you touch something and it feels warm or cold, what is that really telling you about that substance? Temperature is a measure of how fast atoms and molecules are moving in a substance, whether that be the air, a stove top, or an ice cube. The faster those atoms and molecules move, the higher the temperature.

In the metric system, temperature is measured using the Celsius (°C) scale. Because temperature is a condition of the physical properties of a substance, the Celsius scale was created with 100 degrees separating the point at which water freezes, 0 °C, and the point at which water boils, 100 °C. Scientifically, these are the points at which water molecules change from one state of matter to another—from solid (ice) to liquid (water) to gas (water vapor).

Checkpoint

When reading temperatures, it’s important to look beyond the degree symbol to determine which temperature scale the units express. For example, 13 °C reads “13 degrees Celsius,” indicating that the temperature is expressed using the Celsius scale, while 13 °F reads “13 degrees Fahrenheit,” indicating that the temperature is expressed using the Fahrenheit scale.

Who Knew?

How Many Temperature Scales Are There?

Did you know that in addition to Fahrenheit and Celsius, there is a third temperature scale widely used throughout the world? The Kelvin scale starts at absolute zero, the lowest possible temperature at which there is no heat energy present at all. It is primarily used by scientists to measure very high or very low temperatures when water is not involved.

Converting Between Fahrenheit and Celsius Temperatures

Understanding how to convert between Fahrenheit and Celsius temperatures is an essential skill in understanding metric temperatures. You likely know that below 32 °F means freezing temperatures and perhaps that the same holds true for 0 °C. While it may be difficult to recall that water boils at 212 °F, knowing that it boils at 100 °C is a fairly easy thing to remember.

But what about all the temperatures in between? What is the temperature in degrees Celsisus on a scorching summer day? What about a cool autumn afternoon? If a recipe instructs you to preheat the oven to 350 °F, what Celsius temperature do you set the oven at?

Figure 9.16 lists common temperatures on both scales, because we don’t use Celsius temperatures daily it’s difficult to remember them. Fortunately, we don’t have to. Instead, we can convert temperatures from Fahrenheit to Celsius and from Celsius to Fahrenheit using a simple algebraic expression.

An illustration of a thermometer shows temperature in both Fahrenheit and Celsius.
Figure 9.16 Common Temperatures

FORMULA

The formulas used to convert temperatures from Fahrenheit to Celsius or from Celsius to Fahrenheit are outlined in Table 9.3.

Fahrenheit to Celsius Celsius to Fahrenheit
C=59(F−32)C=59(F−32) F=95C+32F=95C+32
Table 9.3 Temperature Conversion Formulas

Example 9.40

Converting Temperatures from Fahrenheit to Celsius

A recipe calls for the oven to be set to 392 °F. What is the temperature in Celsius?

Your Turn 9.40

1.
What is 482 °F in Celsius?

Example 9.41

Converting Temperatures from Celsius to Fahrenheit

On a sunny afternoon in May, the temperature in London was 20 °C. What was the temperature in degrees Fahrenheit?

Your Turn 9.41

1.
What is 75 °C in Fahrenheit?

Example 9.42

Comparing Temperatures in Celsius and Fahrenheit

A manufacturer requires a vaccine to be stored in a refrigerator at temperatures between 36 °F and 46 °F. The refrigerator in the local pharmacy cools to 3 °C. Can the vaccine be stored safely in the pharmacy’s refrigerator?

Your Turn 9.42

1.
In July, the average temperature in Madrid is 16.7 °C. The average temperature in Toronto in July is 57.4 °F. Which city has the higher temperature, and by how many degrees Celsius?

Reasonable Values for Temperature

While knowing the exact temperature is important in most cases, sometimes an approximation will do. When trying to assess the reasonableness of values for temperature, there is a quicker way to convert temperatures for an approximation using mental math. These simpler formulas are listed in Table 9.4.

FORMULA

The formulas used to estimate temperatures from Fahrenheit to Celsius or from Celsius to Fahrenheit are outlined in Table 9.4.

Fahrenheit to Celsius Celsius to Fahrenheit
C=F−302C=F−302 F=2C+30F=2C+30
Table 9.4 Estimate Temperature Conversion

Example 9.43

Using Benchmark Temperatures to Determine Reasonable Values for Temperatures

Which is the more reasonable value for the temperature of a freezer?

  • 5 °C or
  • –5 °C?

Your Turn 9.43

1.
To make candy apples, you must boil the sugar mixture for about 20 minutes. Which is the more reasonable value for the temperature of the mixture when boiling:
  • 48 °C or
  • 148 °C?

Example 9.44

Using Estimation to Determine Reasonable Values for Temperatures

The average body temperature is generally accepted as 98.6 °F. What is a reasonable value for the average body temperature in degrees Celsius:

  • 98.6 °C,
  • 64.3 °C, or
  • 34.3 °C?

Your Turn 9.44

1.
Which represents a reasonable value for temperature of a hot summer day:
  • 102 °C,
  • 37 °C, or
  • 20 °C?

Example 9.45

Using Conversion to Determine Reasonable Values for Temperatures

Which is a reasonable temperature for storing chocolate:

  • 28 °C,
  • 18 °C, or
  • 2 °C?

Your Turn 9.45

1.
Which represents a reasonable temperature for cooking chili:
  • 240 °C,
  • 60 °C, or
  • 6 °C?

Solving Application Problems Involving Temperature

Whether traveling abroad or working in a clinical laboratory, knowing how to solve problems involving temperature is an important skill to have. Many food labels express sizes in both ounces and grams. Most rulers and tape measures are two-sided with one side marked in inches and feet and the other in centimeters and meters. And while many thermometers have both Fahrenheit and Celsius scales, it really isn’t practical to pull out a thermometer when cooking a recipe that uses metric units. Let’s review at few instances where knowing how to fluently use the Celsius scale helps solve problems.

Example 9.46

Using Subtraction to Solve Temperature Problems

The temperature in the refrigerator is 4 °C. The temperature in the freezer is 21 °C lower. What is the temperature in the freezer?

Your Turn 9.46

1.
At 6 PM, the temperature was 4 °C. By 6 AM the temperature had fallen by 6 °C. What was the temperature at 6 AM?

Example 9.47

Using Addition to Solve Temperature Problems

A scientist was using a liquid that was 35 °C. They needed to heat the liquid to raise the temperature by 6 °C. What was the temperature after the scientist heated it?

Your Turn 9.47

1.
A hot dog was cooked in a microwave oven. The hot dog was 4 °C when it was put in the microwave and the temperature increased by 59 °C when it was taken out. What was the temperature of the cooked hot dog?

Example 9.48

Solving Complex Temperature Problems

The optimum temperature for a chemical compound to develop its unique properties is 392 °F. When the heating process begins, the temperature of the compound is 20 °C. For safety purposes the compound can only be heated 9 °C every 15 minutes. How long until the compound reaches its optimum temperature?

Your Turn 9.48

1.
After reaching a temperature of 302 °F a chemical compound cools at the rate of 5 °C every 6 minutes. How long will it take until the compound has reached a temperature of zero degree Celsius.

Check Your Understanding

For the following exercises, determine the most reasonable value for each temperature.
34.
Popsicle:
28.5 °C or
28.5 °F
35.
Room temperature:
20 °F or
20 °C
36.
Hot coffee:
71.1 °C or
71.1 °F
For the following exercises, use mental math to approximate the temperature.
37.
450 °F = __________ °C
38.
35 °C = __________ °F
39.
100 °F = __________ °C
For the following exercises, convert the temperatures to the nearest degree.
40.
225 °F = __________ °C
41.
27 °C = __________ °F
42.
750 °F = __________ °C

Section 9.5 Exercises

For the following exercises, determine the most reasonable value for each temperature.
1.
Human body:
37 °C or
37 °F
2.
An ice cube:
32 °C or
0 °C
3.
Boiling water:
212 °C
or 212 °F
4.
Summer day:
8 °C,
23 °C, or
75 °C
5.
Winter day:
33 °C,
30 °C, or
3 °C
6.
Lava:
700 °C,
70 °C, or
7 °C
For the following exercises, use mental math to approximate the temperature.
7.
500 °F = __________ °C
8.
25 °C = __________ °F
9.
350 °F = __________ °C
10.
150 °C = __________ °F
11.
–4 °F = __________ °C
12.
73 °C = __________ °F
13.
72 °F = __________ °C
14.
10 °C = __________ °F
15.
1,020 °F = __________ °C
For the following exercises, convert the temperatures to the nearest degree.
16.
450 °F = __________ °C
17.
35 °C = __________ °F
18.
525 °F = __________ °C
19.
140 °C = __________ °F
20.
–40 °F = __________ °C
21.
67 °C = __________ °F
22.
85 °F = __________ °C
23.
15 °C = __________ °F
24.
1,200 °F = __________ °C
25.
112 °F = __________ °C
26.
105 °C = __________ °F
27.
125 °F = __________ °C
28.
A cup of tea is 30 °C. After adding ice, the temperature of the tea decreased 3 °C. What is the temperature of the tea now?
29.
A liquid with a temperature of 15 °C is placed on a stovetop. The liquid is heated at the rate of 1.5 °C per minute. What is the temperature of the liquid after 10 minutes?
30.
The temperature inside a store is 22 °C. Outside the temperature is 37 °C. How much cooler is it in the store than outside?
31.
A boiling pan of water is 212 °F. As the water cools the temperature drops 6 °C every 2 minutes. How many minutes until the temperature of the water reaches 40 °C?
32.
On May 1 the temperature was 79 °F. On June 1 the temperature was 25 °C. Which day was warmer?
33.
For each log placed on a fire, the temperature increases 45 °C. How many logs are needed for the campfire to increase 315 °C?
34.
The temperature of a Bunsen burner flame increases 572 °F each minute. About how many minutes does it take for the flame to increase 1,200 °C?
35.
The instructions to cook a pizza say to set the oven at 425 °F. To the nearest degree, what is the temperature in degrees Celsius?
36.
In the evening the temperature was 3 °C. By morning, the temperature had fallen 4 °C. What is the temperature now?
37.
An 8,000 BTU air conditioner can cool a room 1 °C every 5 minutes. If the temperature in the room is 24 °C, how long will it take the air conditioner to cool the room to 20 °C?
38.
What is 95 °F in degrees Celsius?
39.
What is 50 °C in degrees Fahrenheit?
40.
Which is a reasonable value for the temperature of a room:
  • 60 °C,
  • 40 °C, or
  • 20 °C?
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