Lesson 7.3: Scale Factors & Real-World Applications Word Problems

Lesson Description

This lesson focuses on applying scale factors and similarity relationships to real-world word problems. Students will solve problems involving maps, blueprints, models, shadow measurements, and proportional reasoning.

Learning Objectives

• Apply scale factors to determine unknown dimensions.
• Set up and solve proportions from similar figures.
• Solve multi-step real-world geometry word problems.

Topics Covered

• Scale factor definition and interpretation
• Enlargements and reductions
• Map and blueprint scales
• Modeling with similar figures
• Shadow and indirect measurement applications

Key Terms

• Scale Factor: Ratio comparing corresponding side lengths of similar figures.
• Enlargement: A dilation with scale factor greater than 1.
• Reduction: A dilation with scale factor between 0 and 1.
• Proportion: Equation stating two ratios are equal.

Lesson Examples

Example 1: Map Scale

A map uses a scale of 1 inch = 5 miles. Two cities are 3.5 inches apart on the map. How far apart are they in reality?

\[ 3.5 \times 5 = 17.5 \text{ miles} \]

Example 2: Blueprint Reduction

A blueprint uses a scale factor of \( \frac{1}{20} \). If a wall is 40 feet long in reality, how long is it on the blueprint?

\[ 40 \times \frac{1}{20} = 2 \text{ feet} \]

Example 3: Similar Triangles (Shadow Problem)

A 6 ft person casts a 4 ft shadow. A tree casts a 20 ft shadow at the same time. Find the height of the tree.

\[ \frac{6}{4} = \frac{h}{20} \]

\[ 4h = 120 \Rightarrow h = 30 \text{ ft} \]

Practice Problems

1. A model car is built at a scale factor of 1:18. If the real car is 180 inches long, how long is the model?
2. A photograph is enlarged by a scale factor of 1.5. If the original width is 8 inches, what is the new width?
3. A 5 ft pole casts a 3 ft shadow. A nearby building casts a 27 ft shadow. Find the building’s height.

Tips & Common Mistakes

• Keep scale ratios consistent (map to real or real to map).
• Clearly identify corresponding sides before forming proportions.
• Watch units carefully—convert when necessary.

Summary

Scale factors allow us to model, enlarge, and reduce figures accurately. By setting up correct proportions and identifying corresponding measurements, real-world geometry problems become systematic and solvable.

Challenge Problems

1. A rectangular field is represented on a map by dimensions 4 cm by 7 cm. If the scale is 1 cm = 25 m, find the real area of the field.
2. A triangular park is reduced by a scale factor of 0.4. How does its area change compared to the original?
3. A lighthouse casts a 120 ft shadow while a 10 ft pole casts a 8 ft shadow. Determine the lighthouse height.