There are about $1.6 \times 10^3$ meters in a mile. About how many miles are there in $3.52 \times 10^7$ meters? Write your answer in scientific notation, expressed to the exact decimal place.
To determine the number of miles in a given distance of \(3.52 \times 10^7\) meters, we use the conversion factor provided: one mile is equal to approximately \(1.6 \times 10^3\) meters. We seek to find the distance in miles, which we can represent as:
\[\text{Miles} = \frac{\text{Distance in meters}}{\text{Meters per mile}} = \frac{3.52 \times 10^7}{1.6 \times 10^3}.\]
To perform the calculation \(\frac{3.52 \times 10^7}{1.6 \times 10^3}\), we can divide the coefficients and the powers of ten separately. First, calculate the division of the coefficients:
\[\frac{3.52}{1.6} = 2.2.\]
Next, divide the powers of ten using the exponent rule \(\frac{10^m}{10^n} = 10^{m-n}\):
\[\frac{10^7}{10^3} = 10^{7-3} = 10^4.\]
Combining the results from the previous step, we obtain:
\[2.2 \times 10^4.\]
This result is in scientific notation and represents the total number of miles. Thus, there are \(2.2 \times 10^4\) miles in \(3.52 \times 10^7\) meters. The final answer is \(\boxed{2.2 \times 10^4}\).
In navigation and geography, converting units between meters, kilometers, and miles is essential. For example, if you are tracking the distance of a long-range flight or a deep-space transmission signal, the distances are often measured in meters or kilometers, while international navigation standards often use nautical miles or standard miles. Knowing how to use scientific notation for these large conversions allows for efficient and accurate communication of planetary-scale data.