- Convert the distance: \(2.54 \times 10^{-2} \text{ m} = \boxed{2.54 \times 10^{-5} \text{ km}}\).
- Determine leaves: \(300 \text{ pages} / 2 = \boxed{150 \text{ leaves}}\).
- Calculate leaf thickness: \((15 \text{ mm} / 150) = 0.1 \text{ mm} = \boxed{1 \times 10^{-4} \text{ m}}\).
Explanation
- Conversion to Kilometers
The problem asks us to perform unit conversions and calculations involving powers of ten and standard form. First, we convert the distance \(2.54 \times 10^{-2} \text{ m}\) to kilometers. Since \(1 \text{ km} = 10^3 \text{ m}\), it follows that \(1 \text{ m} = 10^{-3} \text{ km}\). Therefore, we multiply the given distance by \(10^{-3}\):
\[2.54 \times 10^{-2} \text{ m} = 2.54 \times 10^{-2} \times 10^{-3} \text{ km} = 2.54 \times 10^{-5} \text{ km}.\]
This is already in the standard form \(a \times 10^n\) where \(1 \le |a| < 10\).
- Determining the Number of Leaves
A book with 300 pages is made up of sheets of paper, where each leaf corresponds to two pages (a front and a back). To find the number of leaves (thicknesses of paper), we divide the total number of pages by 2:
\[\text{Number of leaves} = \frac{300 \text{ pages}}{2 \text{ pages/leaf}} = 150 \text{ leaves}.\]
Thus, 150 leaves make up 300 pages.
- Calculating Leaf Thickness in Meters
Given that the total thickness of the book is \(15 \text{ mm}\) and it consists of 150 leaves, we can calculate the thickness of a single leaf in millimeters:
\[\text{Thickness of 1 leaf} = \frac{15 \text{ mm}}{150 \text{ leaves}} = 0.1 \text{ mm/leaf}.\]
Next, we convert this thickness from millimeters to meters. Since \(1 \text{ mm} = 10^{-3} \text{ m}\), we have:
\[0.1 \text{ mm} = 0.1 \times 10^{-3} \text{ m} = 10^{-1} \times 10^{-3} \text{ m} = 10^{-4} \text{ m}.\]
Expressing this in the required standard form \(a \times 10^n\), we get \(1 \times 10^{-4} \text{ m}\).
Examples
Understanding unit conversion and scientific notation is vital in fields like printing and manufacturing. For example, when an engineer designs the spine of a book, they must know the thickness of the paper used to ensure the final book fits perfectly within its cover. Similarly, scaling distances from meters to kilometers or millimeters to meters is a fundamental skill in map reading, construction, and scientific research, allowing us to manage dimensions ranging from tiny paper fibers to vast geographic expanses.