Applying Similarity and Enlargement in Real-Life Contexts

By the end of this lesson, learners should be able to

1. Calculate missing lengths, surface areas, and capacities of promotional models using scale factors.
2. Determine land areas and masses of similar physical structures using linear, area, and volume scale relationships.

Key vocabulary

Area Scale Factor

The ratio of the area of an image or model to its corresponding original area.

Squaring the linear scale factor gives the area scale factor for the display model.

Volume Scale Factor

The ratio of the volume or capacity of a model to the original volume.

Cubing the linear scale factor yields the volume scale factor for the water container.

Mass Scale Factor

The ratio of the mass of a model solid to its similar original solid.

For identical material density, the mass scale factor equals the volume scale factor.

Lesson preview

Slide 1 · Scale Factors in Marketing Models and ContainersModel · 18 min

When real-life objects such as bottles, containers, or buildings are scaled to create models, the spatial relationships follow exact mathematical power laws. The area scale factor is the square of the linear scale factor, while the volume scale factor is the cube of the linear scale factor. We use these ratios to derive every solution when comparing marketing models or prototypes to their full-sized actual counterparts.

• The linear scale factor is the ratio of corresponding linear dimensions like height or radius.
• The area scale factor equals the square of the linear scale factor for base or surface areas.
• The volume scale factor equals the cube of the linear scale factor for capacity calculations.
• Taking square roots or cube roots allows conversion from area or volume ratios back to linear ratios.

Worked example. Find the linear scale factor (LSF): $\text{ASF}=25$, so $\text{LSF}=\sqrt{25}=5$. Calculate the height of the display replica: $\text{Height}=5\times 20\text{ cm}=100\text{ cm}=1\text{ m}$. Find the volume scale factor (VSF): $\text{VSF}=5^3=125$. Calculate the capacity of the display replica: $\text{Capacity}=125\times 0.5\text{ litres}=62.5\text{ litres}$. Answer: Height of display replica = $100\text{ cm}$ ($1\text{ m}$); Capacity of display replica = $62.5\text{ litres}$..

Check question: A mini model of a cylindrical grain silo has a base area of $0.04{\text{ m}}^2$, while the actual silo has a base area of $36{\text{ m}}^2$. If the model holds $0.016{\text{ m}}^3$ of grain, what is the capacity of the actual silo in ${\text{m}}^3$?

1. $432$
2. $14.4$
3. $1.44$
4. $0.48$

Show answer

$432$. The area scale factor is $36÷0.04=900$. The linear scale factor is $\sqrt{900}=30$. The volume scale factor is ${30}^3=27000$. The actual capacity is $27000\times 0.016=432$.

Slide 2 · Solving Real-World Problems: Fields, Solids, and MassPractice · 22 min

Teach the full lesson on the projector

This lesson, and every other Grade 1–10 CBE lesson, comes ready to project with the EAP ClassKit. It runs without electricity or internet.

• Both slides with diagrams and class activities
• Teacher notes, SNE adaptations and assessment checks
• Homework and the other 10 lessons in this sub-strand

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All 11 lessons in Similarity and Enlargement

1. 1Determining Centre of Enlargement and Linear Scale FactorDetermine the centre of enlargement and the linear scale factor for similar figures
2. 2Finding the Central Point of Dilation and Scale FactorsDetermine the centre of enlargement and the linear scale factor for similar figures
3. 3Constructing Images under Enlargement with Positive and Negative Scale FactorsConstruct the image of an object under an enlargement given the centre of enlargement and the linear scale factor (L.S.F)
4. 4Constructing Images under Geometric EnlargementConstruct the image of an object under an enlargement given the centre of enlargement and the linear scale factor (L.S.F)
5. 5Calculating Area and Volume Scale Factors for Similar FiguresDetermine the area and volume scale factor of different figures and solids
6. 6Determining Area and Volume Scale FactorsDetermine the area and volume scale factor of different figures and solids
7. 7Connecting Length, Surface Area, and Volume RatiosRelate linear scale factor, area scale factor, and volume scale factor in enlargements
8. 8Connecting Linear, Surface Area, and Volumetric Scale Factors in Similar SolidsRelate linear scale factor, area scale factor, and volume scale factor in enlargements
9. 9Applications of Similarity and Enlargement in Real LifeApply similarity and enlargement to real-life situations
10. 10Applying Similarity and Enlargement in Real-Life ContextsApply similarity and enlargement to real-life situations
11. 11Appreciating Similarity and Enlargement in Real-Life ApplicationsAppreciate the use of similarity and enlargement in real-life situations