Lesson 5

Lesson 5: Scaling, Volume, and Capacity

 

Focus

 

Renting a truck to move your belongings can be tricky! If the truck is too big, you will have wasted money and fuel, and the extra room might allow your furniture and boxes to move around too much as you drive. If the truck is too small, you will waste time and fuel making multiple trips.

 

Many rental moving trucks are referred to by their lengths—a 14-ft, 17-ft, or 24-ft truck. In most cases, the height and width of these trucks are the same: only their lengths are different. How does this change in length affect the capacity of each truck?

 

Throughout Lesson 5 you will look at how volume and capacity are affected by changes in linear measurements.

 

Lesson Question

 

In this lesson you will investigate the following question:

• How does altering an object’s linear dimensions affect the object’s volume and capacity?

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 and the Formula Sheet
  
  
• Lesson 5 Assignment (Save a copy of your lesson assignment to your course folder now.)
  
  
• the Project Connection

Materials and Equipment

• calculator

Discover

 

In Try This 1 you will discover the relationship between linear dimensions—length, width, and height—and corresponding changes in volume. You have previously used Exploring Surface Area, Volume, and Nets (Object Interactive). In Try This 1 you will use the applet again to see how volume is affected when you change one, two, or all three dimensions by a linear scale factor.

 

Try This 1

 

Use Exploring Surface Area, Volume, and Nets (Object Interactive) to answer the following questions. Note that in each case of Rectangular Prism, the starting point is a single cube. When the scale factor is 1, the dimensions (length, width, and height) are all 1.

1. Complete a table like the following. Some entries are filled in as examples.
   
   

   Scale Factor	Volume (in cm3) when the following dimension is multiplied by the linear scale factor   • only length	Volume (in cm3) when the following dimensions are multiplied by the linear scale factor   • length • width	Volume (in cm3) when the following dimensions are multiplied by the linear scale factor   • length • width • height
   1 (original cube)	1	1	1
   2	2	4	8
   3
   4
   k

If you need help answering the following questions, complete questions 1 and 2 in Are You Ready? This will help you see the pattern in the table.

2. How is the volume affected if only the length is changed by a factor of k?
   
   
3. How is the volume affected if both length and width are changed by a factor of k?
   
   
4. How is the volume affected if all three dimensions are changed by a factor of k?

Share 1

Share and discuss your responses to the questions in Try This 1 with a classmate or with a group of people. Summarize your discussions by answering the following question:

• What is the relationship between changes in linear dimensions and the effects of the changes on volume?

Save your responses to Try This 1 and a short summary of your discussion from Share 1 in your course folder.

 

Explore

 

Architects need to know the effect of scale factor on volume when they make scale models of buildings. When doubling the dimensions of a flowerbed, gardeners need to know the effect of the scale factor on the flowerbed’s capacity in order to purchase the correct amount of soil.

 

In Discover you examined the relationship between the scale factor—the number by which each dimension is multiplied—and the volume of a cube. You may have discovered this information:

• If one dimension changed by a factor of k, the volume changes by a factor of k.

• If two dimensions are changed by the same scale factor of k, the volume changes by a factor of k2.

• If all three dimensions are changed by the same scale factor of k, the volume changes by a factor of k3.

So, if you multiply each dimension of a cube, for instance, by a factor of 5, the volume of the new cube would be 53, or 125 times, more than the original cube.

 

 

Does this relationship apply to all three-dimensional figures?

 

Try This 2

 

Open Investigating the Volume of Cylinders to help you answer the following questions. When the scale factor is 1, the cylinder radius and height are both equal to 1 m.

 

1. Complete the following table. Some entries are filled in as examples.
   
   

   Scale Factor	Volume (in cm3) when   • only height   is multiplied by the linear scale factor	Volume (in cm3) when   • only radius   is multiplied by the linear scale factor	Volume (in cm3) when   • height • radius   are multiplied by the linear scale factor
   1 (original cylinder)	1	1	1
   2
   3
   4
   5
   k

   
   
2. How is the volume affected if only the height is multiplied by a scale factor of k?
   
   
3. How is the volume affected if only the radius is multiplied by a scale factor of k?
   
   
4. How is the volume affected if both the radius and height are multiplied by the scale factor of k?
   
   

5. The formula for the volume of a cylinder is . If the cylinder height is increased to 2h but the radius remains the same, the new volume becomes 2V.
   
   

   That fact can be shown as follows:
   
   
    
   
   
   
    
   
   
   Show algebraically that, if the radius doubles, the new volume is 4 times the volume of the original.

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

 

Share 2

 

Share your responses to the questions in Try This 2 with a classmate or with a group of people. Then discuss the following:

• What is the relationship between changes in the linear dimensions of cylinders and the effect of the changes on volume?
  
  
• How does this relationship compare to the relationship discussed in Share 1?

If required, save a copy of your discussion in your course folder.