Warm Up The scale of a drawing is 4 in. = 12 ft. Find each actual measurement. 1. 6 in. 2. 2.5 in. The scale of a map is 1 in. = 3.5 mi. Find each length on the map. 3. 21 mi 4. 1.75 mi 18 ft 7.5 ft 6 in. 0.5 in.

Scale Models 7.8 Pre-Algebra

Learn to make comparisons between and find dimensions of scale models and actual objects.

Vocabulary scale model

Scale Model Very large and very small objects are often modeled. A scale model is a three-dimensional model that accurately represents a solid object. The scale model is mathematically similar to the solid object. A scale gives the ratio of the dimensions of the model to the actual dimensions.

Example: Analyzing and Classifying Scale Factors Tell whether each scale reduces, enlarges, or preserves the size of the actual object. A. 1 in:1 yd 1 in. 1 yd 1 in. 36 in. 1 36 = = Convert: 1 yd = 36 in. Simplify. The scale reduces the size of the actual object by a factor of . 1 36

Example: Analyzing and Classifying Scale Factors Tell whether each scale reduces, enlarges, or preserves the size of the actual object. B. 1 m:10 cm 1 m 10 cm 100 cm 10 cm Convert: 1 m = 100 cm. Simplify. = = 10 The scale enlarges the size of the actual object 10 times.

Try This Tell whether each scale reduces, enlarges, or preserves the size of the actual object. A. 1 in:1 ft 1 in. 1 ft 1 in. 12 in. 1 12 Convert: 1 ft = 12 in. Simplify. = = The scale reduces the size of the actual object by a factor of . 1 12

Try This Tell whether each scale reduces, enlarges, or preserves the size of the actual object. B. 12 in:1 ft 12 in. 1 ft 1 ft = = 1 Convert: 12 in. = 1 ft. Simplify. The scale preserves the size of the actual object.

Example: Finding Scale Factors What scale factor relates a 12 in. scale model to a 6 ft. man? 12 in:6 ft State the scale. 12 in 6 ft 12 in. 72 in. 1 6 Write the scale factor as a ratio and simplify. = = The scale factor is , or 1:6. 1 6

Try This What scale factor relates a 12 in. scale model to a 4 ft. tree? 12 in:4 ft State the scale. 12 in 4 ft 12 in. 48 in. 1 4 Write the scale factor as a ratio and simplify. = = The scale factor is , or 1:4. 1 4

Example: Finding Unknown Dimensions Given Scale Factors A model of 32 ft tall house was made using the scale 3 in:2 ft. What is the height of the model? 3 in. 2 ft = 3 in. 24 in. = 1 in. 8 in. 1 8 = First find the scale factor. The scale factor for the model is . Now set up a proportion. 1 8 1 8 = h in. 384 in. Convert: 32 ft = 384 in. 384 = 8h Cross multiply. 48 = h Solve for the height. The height of the model is 48 in.

Try This A model of 24 ft tall bridge was made using the scale 4 in:2 ft. What is the height of the model? 4 in. 2 ft = 4 in. 24 in. = 1 in. 6 in. 1 6 = First find the scale factor. The scale factor for the model is . Now set up a proportion. 1 6 1 6 = h in. 288 in. Convert: 24 ft = 288 in. 288 = 6h Cross multiply. 48 = h Solve for the height. The height of the model is 48 in.

Example: Life Science Application A DNA model was built using the scale 5 cm: 0.0000001 mm. If the model of the DNA chain is 20 cm long, what is the length of the actual chain? Find the scale factor. 5 cm 0.0000001 mm 50 mm = = 500,000,000 The scale factor for the model is 500,000,000. This means the model is 500 million times larger than the actual chain.

Example Continued 500,000,000 1 20 cm x cm = Set up a proportion. Cross multiply. x = 0.00000004 Solve for the length. The length of the DNA chain is 4  10-8 cm.

Try This A model was built using the scale 2 cm:0.01 mm. If the model is 30 cm long, what is the length of the actual object? Find the scale factor. 2 cm 0.01 mm 20 mm = = 2,000 The scale factor for the model is 2,000. This means the actual object is 2 thousand times larger than the model.

Try This Continued 2,000 1 30 cm x cm = Set up a proportion. Cross multiply. Solve for the length. x = 0.015 The length of the actual object is 0.015 cm.

Lesson Quiz: Part 1 Tell whether each scale reduces, enlarges, or preserves the size of the actual object. 1. 75 ft:40 in 2. 1 mi:1760 yd 3. 400 m:1 km 4. What scale factor was used to build a 5 in. model of a 60 ft statue? enlarges preserves reduces 1:144

Lesson Quiz: Part 2 5. To create a model of the Eustachian tube of the human ear, an audiologist used the scale 1.5 cm = 0.6 mm. If the diameter of the Eustachian tube is 1.8 mm, what is the diameter of the model? 4.5 cm