Nicholas is studying volcanoes. He is trying to determine the height of the volcano by looking at a contour map. Looking at the contour map, Nicholas counts 17 contour lines map scale indicates that a contour interval is equivalent to 400 feet. How can he find the height?

In this concept, you will learn to understand three-dimensional scale models and designs.

### Three-Dimensional Figures

There are many different types of maps and models. If you work on interpreting two-dimensional maps, such as scale drawings and floor plans, you will use a scale to interpret measurements the actual dimensions and the scale dimensions. **Two–dimensional representations** mean that the things being represented can be easily shown in a flat plane.

You can also have a three–dimensional method of display. A **three–dimensional figure** contains the length, width and height or depth. A scale model is a model used to represent a three–dimensional space.

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. You compare the scale in the first ratio and you compare the two lengths in the second ratio.

Let’s look at an example.

Brianna is making a scale model of the White House using the scale \begin{align*}1\ cm = 0.5\ m\end{align*}. If the height of Brianna’s model is 42 cm, what is the height of the actual White House?

First, set up a proportion. The scale in the drawing says that

, therefore the proportion is:Next, write the second ratio. You know the scale length is 42 cm. The unknown length is

.\begin{align*}\frac{1\ cm}{0.5\ m} = \frac{42\ cm}{x\ m} \end{align*}Then, cross-multiply to solve for .

\begin{align*}\begin{array}{rcl} \frac{1}{0.5} &=& \frac{42}{x} \\ 1x &=& 42 \times 0.5 \\ x &=& 21 \end{array}\end{align*}The answer is 21.

The height of the actual White House is 21 m.

When you want to show a map in a three-dimensional way, you 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. When contour lines are spread farther apart, the elevation is not as steep. Where contour lines are bunched close together, the elevation is steeper. Contour lines show the general shape of the **terrain** or land.

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.

Let’s look at an example.

This map below 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.

Next, 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.

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

Then, cross-multiply to solve for

.\begin{align*} \begin{array}{rcl} \frac{1}{40} &=& \frac{10}{x} \\ 1x &=& 10 \times 40 \\ x &=& 400 \end{array}\end{align*}

The answer is 400.

The mountain is 400 feet high.

### Examples

#### Example 1

Earlier, you were given a problem about Nicholas and his volcanoes.

Nicholas is trying to determine the height of the volcano by looking at a contour map. He knows the map scale and the number of contour lines.

First, from the map scale it states that the contour interval is 400 feet. That means that each contour line represents 400 feet of elevation.

Next, write the proportion to find the height of the mountain for the 17 contour lines Nicholas counted.

\begin{align*}\frac{\text{1 line}}{\text{400 feet}} = \frac{\text{17 lines}}{\text{x feet}} \end{align*}Then, cross-multiply to solve for ‘

The answer is 6800.

The volcano is 6800 feet high.

#### Example 2

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

First, set up a proportion. The scale for the model is \begin{align*}\frac{1}{4}\text{ inch}= 1\text{ foot}\end{align*}.

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

Next, cross-multiply to solve for

.\begin{align*}\begin{array}{rcl} \frac{0.25}{1} &=& \frac{x}{150} \\ 1x &=& 0.25 \times 150 \\ x &=& 37.5 \end{array}\end{align*}

The answer is 37.5.

The scale model is 37.5 inches long.

Use the contour lines in the diagram below to answer each question.

#### Example 3

What height would be represented by 5 contour lines?

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.

Next, write the proportion to find the height of 5 contour lines.

\begin{align*}\frac{1 \text{ line}}{40 \text{ feet}} = \frac{5 \text{ lines}}{x \text{ feet}} \end{align*}

Then, cross-multiply to solve for

.\begin{align*}\begin{array}{rcl} \frac{1}{40} &=& \frac{5}{x} \\ 1x &=& 5 \times 40 \\ x &=& 200 \end{array}\end{align*}The answer is 200.

The height is 200 feet.

#### Example 4

What height would be represented by 8 contour lines?

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.

Next, write the proportion to find the height of 8 contour lines.

\begin{align*}\frac{1 \text{ line}}{40 \text{ feet}} = \frac{8 \text{ lines}}{x \text{ feet}} \end{align*}

Then, cross-multiply to solve for

.\begin{align*}\begin{array}{rcl} \frac{1}{40} &=& \frac{8}{x} \\ 1x &=& 8 \times 40 \\ x &=& 320 \end{array}\end{align*}

The answer is 320.

The height is 320 feet.

#### Example 5

What height would be represented by 3 contour lines?

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.

Next, write the proportion to find the height of 3 contour lines.

\begin{align*}\frac{1 \text{ line}}{40 \text{ feet}} = \frac{3 \text{ lines}}{x \text{ feet}} \end{align*}

Then, cross-multiply to solve for

.

The answer is 120.

The height is 120 feet.

### Review

Kevin built a scale model of a pool. He used the scale \begin{align*}\frac{1}{2} \text{ inch}= 5 \text{ feet}\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.

4. The map distance of the length of the distance across the lake is 1.5 cm. What is the actual distance across the lake?

5. Explain why the contour lines on the map are closer together at some points and farther apart at other points.

Answer each of the following questions as true or false.

6. A topographic map would include lakes and rivers.

7. A two–dimensional map could also be a topographic map.

8. Three–dimensions means including length, width and height.

9. Depending on what you are measuring, height might be replaced by depth.

10. There is a proportional relationship between length and the area of a figure.

11. A two–dimensional map also includes contour lines.

12. Contour lines can be different sizes if there is a different elevation involved.

13. Elevation also means height.

14. A topographic map can be built in three dimensions.

15. Two –dimensional maps and three–dimensional maps both include the same information.

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 4.10.