- Multiply the length \(1.14 \times 10^4 \text{ mm}\) and width \(12.7 \text{ mm}\) to find the product \(14.478 \times 10^4 \text{ mm}^2\).
- Convert the coefficient \(14.478\) to standard scientific notation as \(1.4478 \times 10^1\).
- Combine the powers of 10 to obtain the final area in scientific notation: \(1.4478 \times 10^5 \text{ mm}^2\).
- The correct option is \(\boxed{1.4478 \times 10^5 \text{ mm}^2}\).
Explanation
- Understanding the Problem
We are given the length of a tape as \(1.14 \times 10^4 \text{ mm}\) and its width as \(12.7 \text{ mm}\). The area \(A\) of a rectangle is calculated as the product of its length and width. Therefore, we need to calculate \(A = (1.14 \times 10^4 \text{ mm}) \times (12.7 \text{ mm})\). Our goal is to express the resulting area in scientific notation, which takes the form \(a \times 10^n\), where \(1 \le a < 10\) and \(n\) is an integer.
- Calculating the Product
To find the total area, we multiply the given values:
\[A = (1.14 \times 10^4) \times 12.7\]
We can group the numerical coefficients and the power of ten:
\[A = (1.14 \times 12.7) \times 10^4\]
Performing the multiplication of the decimals, we get:
\[1.14 \times 12.7 = 14.478\]
Substituting this back into our expression for the area, we have:
\[A = 14.478 \times 10^4 \text{ mm}^2\]
- Expressing in Scientific Notation and Conclusion
Scientific notation requires the coefficient \(a\) to be between 1 and 10. Currently, our coefficient is \(14.478\), which is greater than 10. We can rewrite \(14.478\) as:
\[14.478 = 1.4478 \times 10^1\]
Now substitute this back into the expression for \(A\):
\[A = (1.4478 \times 10^1) \times 10^4 \text{ mm}^2\]
Using the laws of exponents, we add the powers of 10:
\[A = 1.4478 \times 10^{1+4} \text{ mm}^2 = 1.4478 \times 10^5 \text{ mm}^2\]
Comparing this result with the given options, we find that the total area is \(1.4478 \times 10^5 \text{ mm}^2\).
Examples
In real-world applications, such as manufacturing or construction, calculating areas of large rolls of material is a common task. For example, if you are ordering a massive roll of wallpaper or floor covering, knowing the total area in a standard scientific format helps in estimating coverage and material requirements efficiently across different scales, ensuring that large numbers are handled accurately and intuitively.