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# 4.5: Scale in Three Dimensions

Difficulty Level: At Grade Created by: CK-12

## Introduction

A New Kind of Map

Josh spent most of his Saturday morning at the library looking at different books on Mount Everest. After completing his drawings of the Mountain, and figuring out the scale for his model, Josh wanted to look at some maps that other people had created of the mountain.

He began looking in books, but most of the maps weren’t drawn in very high detail. Finally, after a lot of searching, he began using the computer.

Right away, Josh discovered this map on a website.

“What did you find?” his sister Karen asked. She had also been at the library writing a book report.

“I found this map. It is called a relief map,” Josh said.

“What a “relief” that you found it!” Karen joked.

“Not really. It is called a relief map because of what is on it. Look,” Josh began to explain all about the map.

Have you ever seen a relief map? Before Josh explains about relief maps, use this lesson to learn all about them. When finished, you will be able to explain the difference between a relief map and a two-dimensional map.

What You Will Learn

By the end of this lesson, you will be able to complete the following skills:

• Read and interpret plans and views of scale models.
• Read and interpret relief features of topographic maps.
• Compare scale relationship of distance, area and volume.
• Solve real-world problems involving three-dimensional scale models, perspective views and relief maps.

Teaching Time

I. Read and Interpret Plans and Views of Scale Models

In our last lesson, we worked on interpreting two-dimensional maps, scale drawings and floor plans. We used a scale to interpret measurements the actual dimensions and the scale dimensions. These were two – dimensional representations, this means that the things being represented could be easily shown in a flat plane.

What happens is something can’t be shown in a two – dimensional way?

When this happens, we have to use a three – dimensional method of display. Whereas a two – dimensional image takes into account length and width, a three – dimensional figure contains the length, width and height or depth.

When we wanted to represent a two – dimensional space, like a map, we used a scale drawing or a scale map. When we want to represent a three-dimensional space, we use a scale model to represent the space. A scale model is a model used to represent a three – dimensional space.

Yes. You can find the actual dimensions of the space in the same way that you did for a scale drawing.

First, let’s think about how we can find the actual dimensions.

In order to find the actual dimensions from a scale model, you can set up and solve a proportion. The scale given in the model is the first ratio. The unknown length and the scale length is the second ratio. We compare the scale in the first ratio and we compare the two lengths in the second ratio.

Let’s look at an example.

Example

Brianna is making a scale model of the White House using the scale 1 cm = 0.5 m. If the height of Brianna’s model is 42 cm, what is the height of the actual White House?

Set up a proportion. Write the scale as a ratio.

1 cm0.5 m\begin{align*}\frac{1 \ cm}{0.5 \ m}\end{align*}

Now write the second ratio, making sure it follows the form of the first ratio.

1 cm0.5 m=42 cmx m\begin{align*}\frac{1 \ cm }{0.5 \ m}=\frac{42 \ cm}{x \ m}\end{align*}

Next cross-multiply and solve for x\begin{align*}x\end{align*}.

(1)xx=42(0.5)=21\begin{align*}(1)x &= 42(0.5)\\ x &= 21\end{align*}

The actual height of the White House is 21 meters.

We can also use proportions to find the measurements of scale models. Let’s look at an example.

Example

Mike is building a scale model of an airplane using the scale 14 inch=1 foot\begin{align*}\frac{1}{4} \ inch = 1 \ foot\end{align*}. If the actual length of the airplane is 150 feet, what will the length of the scale model be?

Set up a proportion. The scale for the model is 14 inch=1 foot\begin{align*}\frac{1}{4} \ inch = 1 \ foot\end{align*}. So set up a ratio using these values: 0.25 inch1 foot\begin{align*}\frac{0.25 \ inch}{1 \ foot}\end{align*}.

Now write the second ratio. You know the actual length is 150 feet. The unknown length is x\begin{align*}x\end{align*}. Make sure that the second ratio follows the form of the first ratio: inches over feet.

0.25 inch1 foot=x inches150 feet\begin{align*}\frac{0.25 \ inch}{1 \ foot}= \frac{x \ inches}{150 \ feet}\end{align*}

Next cross-multiply to solve for x\begin{align*}x\end{align*}.

(1)xx=150(0.25)=37.5\begin{align*}(1)x &= 150(0.25)\\ x &= 37.5\end{align*}

The length of the scale model is 37.5 inches.

Notice how once you understand how proportions work, you can compare and solve proportions when the measurements of different things have a relationship. The actual length has a relationship with the scale length, so they are related to each other. The scale is a comparison between a selected unit of measure and an actual unit of measure. There is a relationship here as well. These two ratios form a proportion and we have the basis for our problem solving.

II. Read and Interpret Relief Features of Topographic Maps

In the last lesson, you looked at two – dimensional maps. You will notice that if you look at a two – dimensional map, that while it is excellent for measuring distances, it isn’t as helpful when measuring mountains or other features. When we want to show a map in a three-dimensional way, we use a topographic map.

A topographic map is a type of map that shows not only the distances on the ground, but also the relief features of the area, such as mountains. The map uses contour lines to show the elevation of the area. Each contour line is a line of equal elevation or height. They show the general shape of the terrain or land. When contour lines are spread farther apart, the elevation is not as steep. Where contour lines are bunched close together, the elevation is steeper.

Topographic maps may also use colors to represent different features. Blue represents water, green represents vegetation, and brown lines represent topographic contours.

You can interpret these maps by using the scale. There will be a scale to show what each distance on the map represents, just like other maps. There will also be a scale to tell you what each contour line represents.

Example

This map shows a mountain from a national park in California. What is the height of the mountain?

First, look at the map scale. It states that the contour interval is 40 feet. That means that each contour line represents 40 feet of elevation.

Count the number of contour lines that make up the mountain. There are 10 contour lines.

Write a proportion to find the height of the mountain.

1 line40 feet=10 linesx feet\begin{align*}\frac{1 \ line}{40 \ feet}=\frac{10 \ lines}{x \ feet}\end{align*}

Now cross-multiply to solve for x\begin{align*}x\end{align*}.

(1)xx=10(40)=400\begin{align*}(1)x &= 10(40)\\ x &= 400\end{align*}

The mountain is 400 feet high.

Example

Using the map from last example, the width of the base of the mountain on the map is 1.5 inches. What is the actual distance of the base of the mountain?

For this problem, you don’t need the contour scale. You need the map scale. Write a proportion. First we write the scale as a ratio. Then we write the scale measurement compared to the actual measurement for the second ratio. Notice that we are looking for the actual measurement, so that is our missing value.

1 inch0.5 mile=1.5 inchesx miles\begin{align*}\frac{1 \ inch}{0.5 \ mile}=\frac{1.5 \ inches}{x \ miles}\end{align*}

Now cross multiply to solve for x\begin{align*}x\end{align*}.

(1)xx=1.5(0.5)=0.75\begin{align*}(1)x &= 1.5(0.5)\\ x &= 0.75\end{align*}

The distance of the base of the mountain is 0.75 mile.

III. Compare Scale Relationships of Distance, Area and Volume

We can compare the scale relationships of distance, area and volume when looking at three – dimensional figures. If you think back to other math classes, you will remember some of these three – dimensional figures such as a prism or a pyramid. When you compare different measurements, you will see the proportional relationships between them. Let’s look at an example involving volume.

Example

Brooke has a scale model of a warehouse. A storage unit is shaped like a rectangular prism and has the dimensions 4 in. by 3 in. by 6 in. If the scale of the model is 0.5 in. = 2 ft, what are the actual dimensions of the storage unit? What is the volume?

First, notice that there are two parts to this problem. The first part is figuring out the actual dimensions given that Brooke has a scale model. The second part is figuring out the volume.

First use a proportion to find the actual dimensions of the storage unit.

Write the scale as the first ratio, and the scale and unknown actual dimension of the storage unit as the second ratio.

0.5 inch2 feet(0.5)x0.5xx=4 inchesx feet0.5 inch2 feet=3 inchesx feet0.5 inch2 feet=6 inchesx feet=4(2)(0.5)x=3(2)  (0.5)x=6(2)=80.5x=60.5x=12=16  x=12  x=24\begin{align*}\frac{0.5 \ inch}{2 \ feet} &= \frac{4 \ inches}{x \ feet} \qquad \frac{0.5 \ inch}{2 \ feet} = \frac{3 \ inches}{x \ feet} \qquad \frac{0.5 \ inch}{2 \ feet} = \frac{6 \ inches}{x \ feet}\\ (0.5)x &= 4(2) \qquad \qquad \quad (0.5)x = 3(2) \qquad \qquad \ \ (0.5)x = 6(2)\\ 0.5x &= 8 \qquad \qquad \qquad \quad 0.5x =6 \qquad \qquad \qquad \quad 0.5x = 12\\ x &= 16 \qquad \qquad \qquad \quad \ \ x = 12 \qquad \qquad \qquad \quad \ \ x = 24\end{align*}

The actual dimensions of the storage unit are 16 feet by 12 feet by 24 feet. This is the length, width and height of the storage unit.

Now that you know the actual dimensions, you can find the volume.

VVA=lwh=(16 feet)(12 feet)(24 feet)=4,608 feet3\begin{align*}V &= lwh\\ V &= (16 \ feet)(12 \ feet)(24 \ feet)\\ A &= 4,608 \ feet^3\end{align*}

The volume of the storage unit is 4,608 cubic feet.

Now think about how the area of the base of the prism relates to the volume of the prism.

AAA=lw=16(12)=192 sq.feet\begin{align*} A &= lw\\ A &= 16(12)\\ A &= 192 \ sq.feet\end{align*}

If we write the volume as a ratio with the area of the base, we will find something very interesting.

4608192\begin{align*}\frac{4608}{192}\end{align*}

Now divide the numerator by the denominator.

The answer is 24 feet. This is the measurement of the height of the prism.

It means that there is a relationship between the area of a three – dimensional figure its height and its volume. The measurements are related and in proportion to one another.

Example

Tim has a cube with a side length of 4 inches. He has a similar cube with dimensions that are twice the first cube. How does the volume of the larger cube compare to the volume of the smaller cube?

First, find the dimensions of the larger cube.

The problem states that the dimensions are twice those of the first cube. That means they are scaled up by a factor of 2. So the side length of the larger cube is 4 inches×2=8 inches\begin{align*}4 \ inches \times 2 = 8 \ inches\end{align*}.

Now find the volume of both cubes and compare.

Volume of smaller cube:

VVV=lwh=(4 inches)(4 inches)(4 inches)=64 inches3\begin{align*}V &= lwh\\ V &= (4 \ inches)(4 \ inches)(4 \ inches)\\ V &= 64 \ inches^3\end{align*}

Volume of larger cube:

VVV=lwh=(8 inches)(8 inches)(8 inches)=512 inches3\begin{align*}V &= lwh\\ V &= (8 \ inches)(8 \ inches)(8 \ inches)\\ V &= 512 \ inches^3\end{align*}

Next compare the two volumes.

You want to know how the volume of the larger cube compares to the volume of the smaller cube.

Write a ratio comparing the two volumes.

512 inches364 inches3=8\begin{align*}\frac{512 \ inches^3}{64 \ inches^3}=8\end{align*}

The volume of the larger cube is 8 times larger than the volume of the smaller cube.

In the previous example, the scale factor that changed the smaller cube to the larger cube was 2, but the volume was 8 times as large. This leads to the following rule:

• The ratio of volumes of similar figures is the cube of the scale factor.

Write this rule down in your notebooks.

IV. Solve Real – World Problems Involving Three – Dimensional Scale Models, Perspective Views and Relief Maps

The problems that we have been working with in this lesson have all been real – world problems. You can see that working with maps, scale drawings and models have many real-life applications. Think about climbing a mountain! If you were going to climb a mountain it would be very helpful to read a topographic map. However, if you only wanted to know the distance from Cleveland to New York City, then a scale map would be most helpful.

Now let’s expand what we have learned to include perspective views. A perspective is a point of view. When you look at a figure from the top or the side or the back, we can say that we are looking at a three – dimensional figure in perspective.

If we wanted to look at a skyscraper from an airplane, our view would be a perspective view. It would look very different from air as opposed to looking at it from the ground view.

Now let’s go back to the problem from the introduction and use what we have learned to answer the question posed at the beginning of the lesson.

## Real-Life Example Completed

A New Kind of Map

Here is the original problem from the introduction. First, reread it and then answer the question. Explain the difference between a two – dimensional map and a relief map.

Josh spent most of his Saturday morning at the library looking at different books on Mount Everest. After completing his drawings of the Mountain, and figuring out the scale for his model, Josh wanted to look at some maps that other people had created of the mountain.

He began looking in books, but most of the maps weren’t drawn in very high detail. Finally, after a lot of searching, he began using the computer.

Right away, Josh discovered this map on a website.

“What did you find?” his sister Karen asked. She had also been at the library writing a book report.

“I found this map. It is called a relief map,” Josh said.

“What a “relief” that you found it!” Karen joked.

“Not really. It is called a relief map because of what is on it. Look,” Josh began to explain all about the map.

Now explain the difference between a two – dimensional map and a relief map.

Solution to Real – Life Problem

A two – dimensional map is created on a flat surface and only shows the dimensions length and width. There aren’t any other three-dimensional features included on the map. When Josh drew the area of the Everest from space in an earlier lesson, he drew it in a two – dimensional way. He only showed the length, width and area in the scale drawing.

A relief map uses a scale just like any other map, but other features are included on the map. A relief map uses different colors and textures to show the contour of the terrain. It also includes bodies of water and other landmarks. In addition, the map shows contour lines which measure the elevation of a natural land mass.

## Vocabulary

Here are the vocabulary words that are found in this lesson.

Two – Dimensional
A figure drawn in two dimensions is only drawn using length and width.
Three – Dimensional
A figure drawn using length, width and height or depth.
Scale Model
a model that represents a three – dimensional space.
Topographic Map
a map that shows distances on the ground, but also relief features of the map such as mountains.
Contour Lines
lines on a map to show elevation. Each contour line represents the same measure of elevation.
Elevation
the measure of height
Terrain
the land

## Time to Practice

Directions: Solve each problem.

Kevin built a scale model of a pool. He used the scale \begin{align*}\frac{1}{2}^{\prime\prime} = 5 \ ft\end{align*}. Use this information to answer the following questions.

1. The width of the pool on the scale model measures 1.5 inches. What is the actual width of the pool?
2. The length of the pool on the scale model measures 2.5 inches. What is the actual length of the pool?
3. The depth of the pool on the scale model measures 0.5 inches. What is the actual volume of the pool.

This is a map of a national park. Use this information to answer the following questions.

1. What is the height of the mountain shown on the map?
2. The map distance of the length of the distance across the lake is 1.5 cm. What is the actual distance across the lake?
3. Explain why the contour lines on the map are closer together at some points and farther apart at other points.

Directions: Solve each problem.

1. A scale model of a sandbox has dimensions 0.5 inch by 3 inches by 4 inches. If the scale of the model is \begin{align*}\frac{1}{4} \ inch = 1 \ foot\end{align*}, what is the volume of the actual sandbox?
2. A cube measures 5 inches on each side. A similar cube has dimensions that are 3 times as large. How does the volume of the larger cube compare to the volume of the smaller cube?
3. A shipping box measures 16 inches by 12 inches by 8 inches. A second box has a similar size but each dimension is \begin{align*}\frac{1}{4}\end{align*} as long. How does the volume of the second box compare to the volume of the first box?
4. Rina’s fish tank has a volume of 8,000 cubic inches. The dimensions of Ava’s fish tank are all \begin{align*}\frac{1}{2}\end{align*} the size of Rina’s. What is the volume of Ava’s fish tank?

Directions: Answer each of the following questions as true or false.

1. A topographic map would include lakes and rivers.
2. A two – dimensional map could also be a topographic map.
3. Three – dimensions means including length, width and height.
4. Depending on what you are measuring, height might be replaced by depth.
5. There is a proportional relationship between length and the area of a figure.
6. A two – dimensional map also includes contour lines.
7. Contour lines can be different sizes if there is a different elevation involved.
8. Elevation also means height.
9. A topographic map can be built in three dimensions.
10. Two – dimensional maps and three – dimensional maps both include the same information.

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