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Problem Analysis and Solution

The provided document focuses on scale drawing and introduces the concept of measuring lengths using appropriate units. Let's break down the examples and solutions step by step.

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#### Section 19.1: Measuring Lengths

This section discusses the importance of choosing the correct units for measuring different lengths, estimating lengths, and understanding potential errors in measurement. The following conversions are provided as a reference:

- \( 1 \, \text{cm} = 10 \, \text{mm} \)
- \( 1 \, \text{m} = 100 \, \text{cm} \)
- \( 1 \, \text{km} = 1000 \, \text{m} \)

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#### Example 1: Choosing the Most Appropriate Unit

The task is to determine the most appropriate unit of length for measuring various objects or distances. The options are:

(a) Your height
(b) The height of a block of flats
(c) The length of your foot
(d) The thickness of your maths book
(e) The distance between your school and the nearest other school

##### Solution:

1. (a) Your height:
- Human height is typically measured in meters (m). For example, an average adult is about 1.6 to 1.8 meters tall.
- Appropriate unit: \( \text{m} \)

2. (b) The height of a block of flats:
- A block of flats is a large structure, often several stories high. Its height could be tens of meters.
- Appropriate unit: \( \text{m} \)

3. (c) The length of your foot:
- The length of a foot is relatively small and is best measured in centimeters (cm). An average foot length is around 25 to 30 cm.
- Appropriate unit: \( \text{cm} \)

4. (d) The thickness of your maths book:
- The thickness of a book is very small and is best measured in millimeters (mm). A typical book might be 10 to 30 mm thick.
- Appropriate unit: \( \text{mm} \)

5. (e) The distance between your school and the nearest other school:
- This distance is likely to be quite large, possibly several kilometers. Using kilometers (km) is appropriate.
- Appropriate unit: \( \text{km} \)

##### Final Answer for Example 1:
\[
\boxed{
\begin{aligned}
&\text{(a)} \, \text{m} \\
&\text{(b)} \, \text{m} \\
&\text{(c)} \, \text{cm} \\
&\text{(d)} \, \text{mm} \\
&\text{(e)} \, \text{km}
\end{aligned}
}
\]

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#### Example 2: Estimating Lengths

The task is to estimate the lengths of the following objects:

(a) A car
(b) The width of your thumbnail
(c) The length of your pen

##### Solution:

1. (a) A car:
- Cars vary in size, but most passenger cars are between 4 and 5 meters long. A reasonable estimate would be around 4 meters.
- Estimated length: \( 4 \, \text{m} \)

2. (b) The width of your thumbnail:
- The width of a thumbnail is very small. It is typically around 1 to 2 centimeters.
- Estimated width: \( 1 \, \text{cm} \)

3. (c) The length of your pen:
- Pens are generally around 15 to 20 centimeters long. A reasonable estimate would be around 15 cm.
- Estimated length: \( 15 \, \text{cm} \)

##### Final Answer for Example 2:
\[
\boxed{
\begin{aligned}
&\text{(a)} \, 4 \, \text{m} \\
&\text{(b)} \, 1 \, \text{cm} \\
&\text{(c)} \, 15 \, \text{cm}
\end{aligned}
}
\]

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Summary of Solutions

1. Example 1:
\[
\boxed{
\begin{aligned}
&\text{(a)} \, \text{m} \\
&\text{(b)} \, \text{m} \\
&\text{(c)} \, \text{cm} \\
&\text{(d)} \, \text{mm} \\
&\text{(e)} \, \text{km}
\end{aligned}
}
\]

2. Example 2:
\[
\boxed{
\begin{aligned}
&\text{(a)} \, 4 \, \text{m} \\
&\text{(b)} \, 1 \, \text{cm} \\
&\text{(c)} \, 15 \, \text{cm}
\end{aligned}
}
\]

These solutions demonstrate how to choose appropriate units for measuring different lengths and how to make reasonable estimates based on common knowledge.