- The distance \(2.54 \times 10^{-2} \text{ m}\) converts to \(2.54 \times 10^{-5} \text{ km}\).
- A 300-page book contains 150 leaves since each leaf has two pages.
- With a total thickness of \(15 \text{ mm}\) (or \(1.5 \times 10^{-2} \text{ m}\)), the thickness per leaf is \(\frac{1.5 \times 10^{-2} \text{ m}}{150} = 1 \times 10^{-4} \text{ m}\).
- Results: Distance \(\boxed{2.54 \times 10^{-5} \text{ km}}\), Number of leaves \(\boxed{150}\), Thickness per leaf \(\boxed{1 \times 10^{-4} \text{ m}}\).
Explanation
- Problem Analysis
We have three distinct tasks:
- Convert the distance \(2.54 \times 10^{-2} \text{ m}\) into kilometres (\(\text{km}\)) in standard form.
- Determine how many leaves (thicknesses of paper) make up 300 pages.
- Calculate the thickness of one leaf in metres in standard form, given that the total thickness of a 300-page book is \(15 \text{ mm}\).
Recall that standard form (scientific notation) is represented as \(a \times 10^n\), where \(1 \leq |a| < 10\) and \(n\) is an integer. Also, recall the conversion factors: \(1 \text{ km} = 10^3 \text{ m}\) and \(1 \text{ m} = 10^3 \text{ mm}\).
- Distance Conversion
To convert \(2.54 \times 10^{-2} \text{ m}\) to \(\text{km}\), we use the conversion \(1 \text{ m} = 10^{-3} \text{ km}\). Multiplying the given distance by this factor:
\[2.54 \times 10^{-2} \text{ m} = (2.54 \times 10^{-2}) \times 10^{-3} \text{ km}.\]
Applying the exponent rules (\(10^a \times 10^b = 10^{a+b}\)), we have:
\[2.54 \times 10^{-2-3} \text{ km} = 2.54 \times 10^{-5} \text{ km}.\]
Thus, the distance is \(2.54 \times 10^{-5} \text{ km}\).
- Calculating the Number of Leaves
A leaf in a book represents one sheet of paper, which has two sides (two pages). Given that the book has 300 pages:
\[\text{Number of leaves} = \frac{\text{Total pages}}{2} = \frac{300}{2} = 150.\]
So, there are 150 leaves in a 300-page book.
- Calculating the Thickness of One Leaf
The total thickness of the book is given as \(15 \text{ mm}\). First, we convert this to metres:
\[15 \text{ mm} = 15 \times 10^{-3} \text{ m}.\]
To find the thickness of one leaf, we divide the total thickness by the number of leaves:
\[\text{Thickness of one leaf} = \frac{15 \times 10^{-3} \text{ m}}{150}.\]
Calculating this, we get:
\[\text{Thickness of one leaf} = \frac{15}{150} \times 10^{-3} \text{ m} = 0.1 \times 10^{-3} \text{ m}.\]
To express this in standard form (\(a \times 10^n\)), we shift the decimal point:
\[0.1 \times 10^{-3} = 1 \times 10^{-1} \times 10^{-3} = 1 \times 10^{-4} \text{ m}.\]
The thickness of one leaf is \(1 \times 10^{-4} \text{ m}\).
Examples
Understanding how to convert units and calculate physical dimensions is vital in engineering and publishing. For example, if you are designing a book, knowing the thickness of a single sheet of paper allows you to accurately calculate the required width of the book's spine, ensuring it fits perfectly on a bookshelf. Similarly, precise unit conversion is crucial in scientific research to maintain consistency when dealing with data across different scales, such as moving from millimetres to kilometres in mapping or material science.