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Lesson 5

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Course: Math 20-1 SS
Book: Lesson 5
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Date: Monday, 15 September 2025, 2:38 PM

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Table of contents

1. Lesson 5

Mathematics 20-1 Module 5

Module 5: Radicals

 

Lesson 5: Modelling Problems with Radicals

 
Focus

 

This is an image of a planet.

Hemera/Thinkstock

 

One complete orbit of a planet around a star represents the planet’s year. The time it takes for one complete orbit (T) can be accurately predicted by the radical equation , where K is a constant and r is the distance the planet is from the star.

 

As you may have noticed while working on Module 5 Project: Plan a Planet, radicals are found in many different equations associated with the universe. Throughout this module you have seen a wide variety of equations that use radicals. In this lesson you will use radical equations to model a variety of different situations on Earth and throughout the universe. You will develop some of these equations yourself.

 

Outcomes

 

At the end of this lesson you will be able to

  • solve a problem that involves radical expressions

  • solve problems by modelling situations using radical equations
Lesson Question

 

You will investigate the following question:

  • How do you solve problems by modelling situations using radical equations?
Assessment


Your assessment may be based on a combination of the following tasks:

  • completion of the Lesson 5 Assignment (Download the Lesson 5 Assignment and save it in your course folder now.)

  • course folder submissions from Try This and Share activities

  • additions to Module 5 Glossary Terms and Formula Sheet

  • work under Project Connection


1.1. Launch

Mathematics 20-1 Module 5

Module 5: Radicals

 

Launch

 

Do you have the background knowledge and skills you need to complete this lesson successfully? This section, which includes Are You Ready? and Refresher, will help you find out.


Before beginning this lesson you should be able to

  • square a radical when the radicand (the number under the root sign) is a binomial
  • square a binomial containing a radical


1.2. Are You Ready?

Mathematics 20-1 Module 5

Module 5: Radicals

 

Are You Ready?

 

Complete the following questions. If you experience difficulty and need help, visit Refresher or contact your teacher.

  1. Square the following terms.

    1. Answer

    2. Answer

    3. Answer

  2. Square the following terms, remembering to use the distributive property as needed and to simplify:

    1. r − 6 Answer

    2. Answer

    3. Answer

    4. Answer

How did the questions go? If you feel comfortable with the concepts covered in the questions, skip forward to Discover. If you experienced difficulties, use the resources in Refresher to review these important concepts before continuing through the lesson.


 

1.3. Refresher

Mathematics 20-1 Module 5

Module 5: Radicals

 

Refresher

 

Review this example, which shows the squaring of a radical with a binomial radicand.

 

Simplify .

Notice that the following two expressions are not equal since the right side is valid for any value of q, but the left side is only valid for .

 

 

 

Squaring a square root is an example of inverse operations, so  becomes 3q − 7, but with a restricted domain—the domain of the original expression, .

This is a play button that opens “Operations with Radical Expressions.”

Screenshot reprinted with permission of ExploreLearning

“Operations with Radical Expressions” helps identify the correct steps in completing operations with radical expressions. Click the “new” button to find examples of multiplication of binomials that contain radicals.


This is a play button that opens “Square a Binomial.”

Khan Academy CC BY-NC-SA 3.0

If you are having difficulty squaring binomials (without radicals) watch “Square a Binomial.”

Go back to the Are You Ready section and try the questions again. If you are still having difficulty, contact your teacher.



1.4. Discover

Mathematics 20-1 Module 5

Module 5: Radicals

 

Discover

 

In this section you will explore the graph of an equation involving a radical.


Try This 1

 

This is a play button that opens Graphing Tool.

  1. Sketch using your graphing calculator or the multimedia applet Graphing Tool. hint

  2. Sketch y = x2 using your graphing calculator or Graphing Tool.

  3. What similarities and differences exist between these two graphs?

  4. Isolate x in .

  5. What do you notice about the form of the equation when x is isolated?

course folder Save your work in your course folder.

 

Share 1

 

Based on your observations from the graphing exercise above, respond to the following questions with a partner or group.

  1. Squaring a radical equation changes the equation into what kind of equation? Is a radical equation half of a parabola turned on its side?

  2. Summarize your dicussion by speculating about the relationship between radical equations and quadratic equations.

course folder Save your work in your course folder.

Graphing Tool Instructions.

 

Open Graphing Tool. Select Graphing Tool Interactive Manual. Watch and listen to Text Entry of Functions (in the menu on the left) to see an overview of how the tool functions. Download the tool that applies to your computer platform, Mac Version or PC Version.

 

After downloading, Graphing Tool will open with an image shown on your computer screen. Click on the image for instructions on how to get started. Use the Functions and Relations tab and choose Create a Text Entry Function to begin entering your equation.

 

1.5. Explore

Mathematics 20-1 Module 5

Module 5: Radicals

 

Explore

 

Over 500 years ago, Galileo discovered that a swinging pendulum kept the same time, whether the pendulum swung in a tiny arc or in a slightly greater arc. Galileo used this understanding to measure the pulse of medical patients. He found that the length of the pendulum was the determining factor in the pendulum’s period of motion.

 

In Try This 2 you will model the pendulum using a radical equation.

 

Try This 2

 

This is a photo of a wall clock with a pendulum.

Hemera/Thinkstock

Pendulum clocks can be adjusted to be accurate timekeepers by carefully adjusting the length of the pendulum.

 

One of the relationships of physics is that the time for a complete back-and-forth swing of a pendulum is

  • directly proportional to the square root of the length
  • inversely proportional to the square root of the acceleration due to gravity

The constant of proportionality is 2π.

 

Another way of describing the relationship is as follows:

  • The time period is 2π times the square root of length divided by the square root of the acceleration due to gravity.
  1. Use Write an Equation to express the relationship between the period T of a pendulum and its length. Use L for length and g for the acceleration due to gravity.

     
    This is a play button that opens Write an Equation.

  2. A pendulum in a wall clock has a length of 1.4 m. What is the time period of that pendulum to the nearest hundredth of a second if the acceleration due to gravity is 9.81 m/s2 at that location? Use Time Period of the Pendulum to find the answer. hint

     
    This is a play button that opens Time Period of the Pendulum.

  3. Rearrange the equation in Isolate L. hint

     
    This is a play button that opens Isolate L.

  4. Find the length of a pendulum that will give a time period of exactly 2 s, so that the clock will tick precisely every second as the pendulum swings back and forth. Record your answer to the nearest hundredth of a second. Use Length of a Pendulum.

     
    This is a play button that opens Length of a Pendulum.
There are a few steps you need to work through to isolate L. You may want to try this on paper first, and then complete Isolate L. Your paper answer may look slightly different than your answer in Isolate L as there are many ways to isolate L. You will also need to use your ability to identify equivalent equations.
It can be helpful to put the units into the equation.

1.6. Explore 2

Mathematics 20-1 Module 5

Module 5: Radicals

 

Try This 3


textbook

Access hints as you need them to complete this activity.

  1. Study “Example 4” on page 299 of the textbook. Note the two ways to solve the equation.

  2. Solve “Your Turn” at the bottom of page 299. It may be helpful to make your own drawing of the bottom of the box and the side view.

    hint hint hint hint
Self-Check 1

 

This photo shoes a young woman drying her hair with a hair dryer.

Photodisc/Thinkstock

  1. The power in watts (W) of an electric appliance is equal to the current in amps (A) squared multiplied by the resistance in ohms (Ω).

    1. Express the statement above as an equation, where power is P, current is I, and resistance is R. Answer

    2. Rearrange the equation to isolate current, I. Answer

    3. How much current flows through a 1600-W hair dryer that has a resistance of 8.82 Ω? Answer
  2. Most new television screens have a 16:9 aspect ratio. This means the width is  times the height. When a television is advertised as having a 42-in screen, it means the length of the diagonal of the screen is 42 in.

    1. What is the height of a 42-in screen? Begin by letting the height be h. Then express the width in terms of the height. Make a drawing using your variables and putting in the length of the diagonal. Use the Pythagorean theorem to set up an equation and solve for the height. Is your answer reasonable? Answer

    2. What is the width of a 42-in screen? How can you check your answer? Answer

    3. What is the total viewing area of a 42-in screen? Answer

Skip forward to Connect if you feel you have a solid understanding of how to

  • solve a problem that involves radical expressions

  • solve problems by modelling situations using radical equations

If you need a bit more practice, complete Self-Check 2.

 

Self-Check 2


textbook

Complete any or all of questions 14, 15, 16, and 17 on pages 301 and 302 of the textbook. As you finish each part of a question, check your work against the answers given at the back of the textbook. If you are still unclear about how to answer some questions, ask your teacher about those questions and get some help.



It is important to check your work here before moving on. You should have . If d = 23.3 cm, A = 181 cm2.
Use the Pythagorean theorem a second time to get the length of the diagonal of the box.
Use the Pythagorean theorem to get the length of the diagonal of the bottom of the box.
The area of one face is A, so the length of one side is .

 

This is an image of a square.

1.7. Connect

Mathematics 20-1 Module 5

Module 5: Radicals

 

Connect

 

Lesson 5 Assignment


assignment

Open your copy of Lesson 5 Assignment, which you saved in your course folder at the beginning of this lesson. Complete the assignment.

 

course folder Save your work in your course folder.

 

Project Connection


assessment

Complete Activity 2 and Conclusion of Module 5 Project: Plan a Planet.

 

course folder Save your work in your course folder.



 

1.8. Lesson 5 Summary

Mathematics 20-1 Module 5

Module 5: Radicals

 

Lesson 5 Summary

 

This image shows a planet and nebulae.

iStockphoto/Thinkstock

In this lesson you investigated the following question:

  • How do you solve problems by modelling situations using radical equations?

You learned to model a variety of situations using radical equations. You learned that making a diagram is often helpful in getting the equation to match the situation.

 

You also learned that equations can be solved even if the variable needed is part of a radicand. Equations can be solved by rearranging and squaring both sides of the equation, or by substituting the values into the equation as it is.