- The distance \(2.54 \times 10^{-2} \text{ m}\) converts to \(2.54 \times 10^{-5} \text{ km}\).
- A 300-page book consists of \(300 / 2 = 150\) leaves.
- The total thickness of 15 mm corresponds to \(0.015 \text{ m}\).
- Dividing the total thickness by 150 leaves yields a leaf thickness of \(1.0 \times 10^{-4} \text{ m}\).
- Final answers: \(2.54 \times 10^{-5} \text{ km}\); \(150 \text{ leaves}\); \(1.0 \times 10^{-4} \text{ m}\). \(\boxed{2.54 \times 10^{-5} \text{ km}, 150, 1.0 \times 10^{-4} \text{ m}}\).
Explanation
- Problem Analysis
First, we analyze the given measurements and properties:
- Distance to convert: \(2.54 \times 10^{-2} \text{ m}\).
- Book properties: Total pages = 300, Total thickness = \(15 \text{ mm}\).
- Definitions:
- A 'leaf' corresponds to a single sheet of paper, which has two sides (two pages).
- Conversion factors: \(1 \text{ km} = 10^3 \text{ m}\) and \(1 \text{ m} = 10^3 \text{ mm}\).
- Standard form: A number expressed as \(a \times 10^n\), where \(1 \le a < 10\) and \(n\) is an integer.
- Conversion of Distance to km
To express \(2.54 \times 10^{-2} \text{ m}\) in kilometers (km), we use the relation \(1 \text{ km} = 10^3 \text{ m}\), which means \(1 \text{ m} = 10^{-3} \text{ km}\).
Multiplying the given distance by this factor:
\[2.54 \times 10^{-2} \text{ m} = (2.54 \times 10^{-2}) \times 10^{-3} \text{ km} = 2.54 \times 10^{-5} \text{ km}.\]
The distance in kilometers is \(2.54 \times 10^{-5} \text{ km}\).
- Determining the Number of Leaves
In a book, one leaf is a single sheet of paper containing two pages (front and back). Since the book has 300 pages numbered 1 to 300, we divide the total number of pages by 2 to find the number of leaves:
\[\text{Number of leaves} = \frac{300 \text{ pages}}{2 \text{ pages/leaf}} = 150 \text{ leaves}.\]
Thus, there are 150 leaves in a 300-page book.
- Calculation of Leaf Thickness
The total thickness of the book is 15 mm. To find the thickness of one leaf in metres, we first convert the total thickness from mm to m:
\[15 \text{ mm} = 15 \times 10^{-3} \text{ m} = 0.015 \text{ m}.\]
Next, divide this total thickness by the number of leaves (150):
\[\text{Thickness of one leaf} = \frac{0.015 \text{ m}}{150} = 0.0001 \text{ m}.\]
To express this in standard form:
\[0.0001 \text{ m} = 1.0 \times 10^{-4} \text{ m}.\]
Therefore, the thickness of one leaf is \(1.0 \times 10^{-4} \text{ m}\).
Examples
Understanding scientific notation and unit conversion is essential in fields like engineering and publishing. For instance, when designing a book, an editor needs to know the exact thickness of paper to estimate the width of the spine. If a book has 500 pages and each leaf is \(0.1 \text{ mm}\) thick, the spine width calculation involves simple division and unit conversion, ensuring the cover fits perfectly. This skill allows us to work accurately with very small or very large dimensions in everyday technical tasks.