Lesson 4

Lesson 4: Applying Volume and Capacity Formulas

 

Focus

 

People wouldn’t survive without water. Water tanks are used around the world to store water and help create enough pressure to move the liquid to each person in the system. As you can see from the photos, water towers can come in a variety of different shapes. Can you identify any cylinders, cones, or spheres in the towers pictured?

 

Most objects are not pure spheres, pyramids, cones, cylinders, or prisms. Instead, objects are often combinations of many shapes. In this lesson you will apply volume and capacity formulas to a variety of practical problems, including composite figures.

 

cylinder tower: Comstock/Thinkstock; round tower: iStockphoto/Thinkstock; cone tower: iStockphoto/Thinkstock

 

Lesson Questions

 

In this lesson you will investigate the following questions:

• How are volume and capacity formulas applied to problems involving composite figures?
  
  
• How is formula manipulation used to find unknown dimensions?

Assessment

Your assessment for this lesson may include a combination of the following:

• course folder submissions from the Try This and Share sections of the lesson
  
  
• your contribution to the Mathematics 20-3 Glossary Terms
  
  
• Lesson 4 Assignment (Save a copy of your lesson assignment to your course folder now.)
  
  
• the Project Connection

Materials and Equipment

• calculator

Launch

 

This section checks to see if you have the necessary background knowledge and skills required to successfully complete Lesson 4.

 

Complete the following Are You Ready? questions. If you have difficulty or any questions, visit Refresher for a review or contact your instructor.

 

Are You Ready?

1. Calculate . Answer
   
   
2. In Lessons 2 and 3 you explored volume formulas for prisms and pyramids, for cylinders and cones, and for spheres. Review all these formulas by working through Volume Formula Review.
   
   
   

If you answered the Are You Ready? questions without problems, move on to Discover.

 

If you found the Are You Ready? questions difficult, complete Refresher to review these topics.

 

Refresher

 

If you don't know the answers to the Are You Ready? questions or require more information, review the following.

 

For a review of the different volume formulas, work through Exploring Surface Area, Volume, and Nets (Object Interactive).

 

 

Go back to Are You Ready? and try the questions again. Contact your teacher if you continue to have difficulty with the questions.

 

Discover

 

 

Landscapers deal with irregular shapes (composite figures) on a regular basis. Flower beds and swimming pools are two such examples.

 

A landscaper knows that calculating volume and capacity are important skills, since understanding these components helps determine the cost and time required to complete projects. For example, in order to install a swimming pool, the landscaper must estimate the volume of soil to be removed to fit the pool in the ground.

 

Try This 1

 

Look at the following custom-designed swimming pool.

 

 

To determine the pool’s volume, you need to divide the composite figure into objects for which you have formulas.  

1. What simple 3-D objects will you divide the swimming pool into? Draw a diagram to show the divisions.
    
2. What formulas will you use to determine the volume of each 3-D object?
   
   
3. What dimensions will you need to calculate the volume of each object?
   
   
4. What referents will you use to estimate those dimensions?
   
   
5. What measuring tools will you need to determine the dimensions?
   
   
6. What imperial units will you pick to calculate the volume?

Share 1

Share your responses to the questions in Try This 1 with a classmate or with a group of people.

• How did your divisions into simple objects compare with the divisions of your classmate? After viewing other divisions, would you change how you divided up the pool? Was there a division method that made the question easier to solve? If so, what was the division?

Save a copy of your responses to Try This 1 in your course folder.

 

Explore

 

In this section you will explore how volume formulas are applied to practical everyday problems. Many of these situations will involve composite figures.

 

Try This 2

 

 

Step 1: Open Composite 3D Figure.

 

Step 2: Read through Example Problem, Volume Solution One, and Volume Solution Two.

 

Step 3: Answer the questions that follow.

1. What is the difference between Volume Solution One and Volume Solution Two?
   
   
2. Which solution did you find easier to understand? Why?

Save your responses from Try This 2 in your course folder.

 

As you read through the following examples, think of different ways in which the composite figures could have been divided.

 

Example

A gelatine capsule containing medication is 1.11 cm in overall length and 0.491 cm in diameter. This capsule is a composite figure made up of a cylinder with two hemispherical ends. What is the capsule’s capacity in millilitres? Round your answer to two decimal places.

 

Note: A hemisphere is half a sphere, and 1 mL = 1 cm3.

 

Creatas Images/Thinkstock

 

Solution

 

The two hemispherical ends form one sphere with a diameter of 0.491 cm.

 

 

 

Creatas Images/Thinkstock

 

The radius, r, of the sphere and of the cylinder is one-half the diameter, d.

 

 

 

Find the height, h, of the central cylinder.

 

 

Creatas Images/Thinkstock

 

 

 

Find the total volume of the capsule.

 

 

 

Because 1 mL = 1 cm3, the capacity of the capsule is about 0.18 mL.

 

Example

 

Consider the pillar shown.

1. How much concrete is needed to make the top piece of the brick pillar?

2. Concrete is sold in cubic yards. Convert the volume to cubic yards. Round your answer to three decimal places.

Solution

1. The total volume of the pillar is made up of the volume of the rectangular (square) prism and the volume of the square pyramid (tetrahedron).
    

The volume of concrete needed is 3024 in3.

2. To convert inches to yards use 1 in = 0.0278 yd.