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Have you ever noticed how a model car or a map looks like a smaller version of the real thing? This happens because of scaling. Scaling is the process of changing the size of an object or quantity while keeping its proportions the same. In other words, every dimension of the original object is multiplied or divided by the same number, called the scale factor.
Why is scaling important? It helps us understand and work with sizes that are too big or too small to measure or see directly. Architects use scaling to create models of buildings, cartographers make maps of large areas, and engineers build miniature prototypes for testing.
In this chapter, we will learn what scaling means mathematically, how to calculate the scale factor, and how scaling affects length, area, and volume. We will explore real-life examples with clear steps and practice problems to prepare you for competitive exams.
The scale factor tells us how much larger or smaller the scaled object is compared to the original. It is the number by which every length in the original object is multiplied to get the length in the scaled object.
If the scale factor k is greater than 1, the object is enlarged (scaled up). If k is less than 1, the object is reduced (scaled down).
For example, consider a square with side length 4 cm. If we apply a scale factor of 2, the new square's sides are:
\[4 \text{ cm} \times 2 = 8 \text{ cm}\]This means the scaled square is twice as large in every dimension.
Notice how the shape remains the same; only the size changes. This is the essence of scaling.
Scaling affects not just the length but also the area and volume of objects. However, the changes in area and volume do not happen at the same rate as length.
Why do these changes differ? Area involves two dimensions (length and width), so each dimension is scaled by \(k\), giving \(k \times k = k^2\). Volume involves three dimensions (length, width, and height), so it scales by \(k \times k \times k = k^3\).
Let's compare these effects on a cube with original side length 1 cm, area of one face 1 cm², and volume 1 cm³.
| Scale Factor (\(k\)) | Scaled Length (cm) | Scaled Area (cm²) | Scaled Volume (cm³) |
|---|---|---|---|
| 1 (original) | 1 | 1 | 1 |
| 2 | 2 | 4 (2²) | 8 (2³) |
| 3 | 3 | 9 (3²) | 27 (3³) |
| 0.5 | 0.5 | 0.25 (0.5²) | 0.125 (0.5³) |
Observe that even a small change in scale factor leads to large changes in volume. This fact is essential when working with models, maps, and real-world measurements.
To find the scale factor when you know two corresponding lengths, use the formula:
Make sure both lengths are in the same unit before dividing.
Many important fields use scaling every day. For example:
Understanding scaling allows you to interpret and create such representations accurately.
Step 1: Identify the formula for scale factor:
\( k = \frac{\text{Scaled Length}}{\text{Original Length}} \)
Step 2: Substitute the given values:
\( k = \frac{15 \, \text{cm}}{5 \, \text{cm}} \)
Step 3: Calculate:
\( k = 3 \)
Answer: The scale factor is 3, indicating the object is scaled up 3 times.
Step 1: Recall that area scales by \( k^2 \).
Step 2: Calculate \( k^2 = 3^2 = 9 \).
Step 3: Multiply original area by \( k^2 \):
\( 30 \times 9 = 270 \, \text{cm}^2 \)
Answer: The scaled rectangle has an area of 270 cm².
Step 1: Find the original volume:
\( V = 4^3 = 64 \, \text{cm}^3 \)
Step 2: The scale factor \( k = \frac{8}{4} = 2 \).
Step 3: Volume scales by \( k^3 \). Calculate \( k^3 = 2^3 = 8 \).
Step 4: New volume is:
\( V' = 8 \times 64 = 512 \, \text{cm}^3 \)
Answer: The new volume is 512 cm³.
Step 1: Understand the scale: 1 cm on the map = 50,000 cm in reality.
Step 2: Calculate actual distance:
\( 4 \, \text{cm on map} \times 50,000 = 200,000 \, \text{cm} \)
Step 3: Convert cm to meters:
\( 200,000 \, \text{cm} = 2000 \, \text{m} \) (since 1 m = 100 cm)
Step 4: Convert meters to kilometers:
\( 2000 \, \text{m} = 2 \, \text{km} \)
Answer: The actual distance is 2 km.
Part (a): Finding actual height
The scale factor \( k = \frac{1}{50} \) means model dimension = \(\frac{1}{50}\) of actual.
Let actual height be \(H\):
\( \frac{1}{50} \times H = 3 \, \text{m} \ \Rightarrow \ H = 3 \times 50 = 150 \, \text{m} \)
Part (b): Finding actual base area
Area scales by \( k^2 = \left(\frac{1}{50}\right)^2 = \frac{1}{2500} \).
Model area = \(\frac{1}{2500}\) of actual area.
Let actual base area be \(A\):
\( \frac{1}{2500} \times A = 4 \, \text{m}^2 \ \Rightarrow \ A = 4 \times 2500 = 10,000 \, \text{m}^2 \)
Part (c): Ratio of volumes
Volume scales by \( k^3 = \left(\frac{1}{50}\right)^3 = \frac{1}{125,000} \).
Thus, actual volume is 125,000 times the model volume.
Answer:
When to use: Scaling area or volume from linear scale factor.
When to use: Before solving scale-related problems.
When to use: At the start of scaling problems.
When to use: Multi-step or difficult scaling questions.
When to use: Problems involving maps and blueprints.
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