Section 7.7Scale Drawing and Models

A scale model or scale drawing is an object or drawingwith lengths proportional to the object it represents. The scale of a model or drawing is the ratio of a length on the model or drawing to the actual length of the object being modeled or drawn.

Example 1:

a) The distance between Boston and Chicago on a map is 9 inches. If the scale of the map is 1 inch: 95 miles, what is the actual distance from Boston to Chicago?

Write and solve a proportion.

Let x represent the distance between Boston and Chicago.

1 in. 9 in.

95mi mixmap →

actual →

← map

← actual

1 ● x =95 ● 9 Cross Products Property

x = 855 miles Simplify.

The distance between Boston and Chicago is 855 miles.

Example 1:

b) The distance between Cheyenne, WY, and Tulsa, OK, on a map is 8 inches. If the scale of the map is 1 inch : 90 miles, what is the actual distance from Cheyenne to Tulsa?

Write and solve a proportion.

Let x represent the distance between Cheyenne and Tulsa.

1 in. 8 in.

90mi mixmap →

actual →

← map

← actual

1 ● x =90 ● 8 Cross Products Property

x = 720 miles Simplify.

The distance between Cheyenne and Tulsa is 720 miles.

The scale factor of a model or drawing is written as a unitless ratio in simplest form. Scale factors are always written so that the model length in the ratio comes first.

Example 2:

a) A miniature replica of a fighter jet is 4 inches long. The actual length of the jet is 12.8 yards. What is the scale of the model? How many times as long as the actual is the model jet?

To find the scale, write the ratio of a replica length to an actual length. replica length 4 in. 1 in.

actual length 12.8 yd 3.2yd

The scale of the model is 1 in. : 3.2 yd.

To answer the second question, find the scale factor of the model. Multiply by a conversion factor that relates inches to yards to obtain a unitless ratio.

The scale factor is 115.2 : 1. That is, the actual jet is 115.2 times as long as the model.

1in. 1in. 1ft 1yd 1

3.2 yd 3.2 yd 12 in. 3 ft 115.2

Example 2:

b) A miniature replica of a fire engine is 9 inches long. The actual length of the fire engine is 13.5 yards. What is the scale of the replica? How many times as long as the model is the actual fire engine?

To find the scale, write the ratio of a replica length to an actual length. replica length 9 in. 2 in.

actual length 13.5 yd 3yd

The scale of the model is 2 in. : 3 yd.

To answer the second question, find the scale factor of the model. Multiply by a conversion factor that relates inches to yards to obtain a unitless ratio.

The scale factor is 54 : 1. That is, the actual fire engine is 54 times as long as the model.

2in. 2 in. 1ft 1yd 2 1

3 yd 3 yd 12 in. 3 ft 108 54

Example 3:

a) Gerrard is making a scale model of his classroom on an 11-by-17 inch sheet of paper. If the classroom is 20 feet by 32 feet, choose an appropriate scale for the drawing and determine the drawing’s dimensions.

The actual classroom is 20 feet wide.20 feet ÷ 11 inches = 1.8 feet per inchThe actual classroom is 32 feet long. 32 feet ÷ 17 inches = 1.8 feet per inch

A scale of 1 inch = 2 feet would be appropriate. So, for every inch on the paper p, let the actual measure a be 2 feet. Write this as an equation.

width:a = 2 ● p Write an equation.20 = 2 ● p Substitute a = 2010 = p Divide each side by 2.

1 in. : 2 ft; 10 in. wide, 16 in. long

length:a = 2 ● p Write an equation.32 = 2 ● p Substitute a = 3216 = p Divide each side by 2.

Example 3:

b) Alaina is an architect making a scale model of a house in a 15-by-26 inch display. If the house is 84 feet by 144 feet, what would be the dimensions of the model using a scale of 1 in. : 6 ft?

Using a scale of 1 inch = 6, for every inch on the paper p, let the actual measure a be 6 feet. Write this as an equation.

14 in. wide, 24 in. long

length:a = 6 ● p Write an equation.144 = 6 ● p Substitute a = 14424 = p Divide each side by 6.

width:a = 6 ● p Write an equation.84 = 6 ● p Substitute a = 8414 = p Divide each side by 6.