Problem 51 - Chemistry: An Atoms-Focused Approach

Question

A single strand of natural silk may be as long as \(4.0 \times 10^{3} \mathrm{m}\) What is this length in miles?

Step-by-Step Solution

Answer

Question: Convert a length of 4.0 x 10^3 meters of silk to miles. Answer: The length of the silk is approximately 2.485 miles.

Step 1: Write down the given length and conversion factor

The given length of silk is \(4.0 \times 10^3 \mathrm{m}\). The conversion factor between meters and miles is 1 mile = 1,609.34 meters.

Step 2: Set up the conversion

We want to convert the length of silk from meters to miles using the conversion factor. To do that, we can set up a proportion like this: $$ \frac{Length\ in\ meters}{Length\ in\ miles} = \frac{1,609.34\ meters}{1\ mile} $$

Step 3: Substitute the known length and solve for the unknown length

Now we can substitute the given length of silk into the equation and solve for the length in miles: $$ \frac{4.0 \times 10^3 \mathrm{m}}{Length\ in\ miles} = \frac{1,609.34\ meters}{1\ mile} $$ To find the length in miles, multiply both sides of the equation by "Length in miles" and then divide by 1,609.34 meters: $$ Length\ in\ miles = \frac{(4.0 \times 10^3 \mathrm{m})(1 \mathrm{mile})}{1,609.34 \mathrm{m}} $$

Step 4: Simplify and calculate the length in miles

Now we simplify the equation and calculate the length of silk in miles: $$ Length\ in\ miles = \frac{4.0 \times 10^3 \mathrm{m}}{1,609.34 \mathrm{m}} $$ $$ Length\ in\ miles = 2.485 \mathrm{miles} $$ So, the length of the silk is approximately 2.485 miles.

Key Concepts

Conversion Factor

Understanding the concept of conversion factors is crucial when transitioning between different units of measurement. A conversion factor is a number used to change one set of units to another, by multiplying or dividing. This number is derived from the relationship between the two units. In our problem, the conversion factor between meters and miles is 1 mile = 1,609.34 meters.
Here is how you effectively use a conversion factor:

• Ensure the conversion factor is in a ratio form that aids your conversion process, such as 1,609.34 meters per 1 mile.
• Multiply or divide the existing measurement by this conversion factor to get the desired unit.

This method allows accurate and straightforward conversions, ensuring that measurements are consistent across different units.

Length Measurement

Length measurement is the determination of something from end to end. It can be measured in various units, depending on the system of measurement being used, such as the metric or imperial systems.
Here are some key points about length measurement:

• The metric system uses meters as the base unit of length. This includes kilometers for long distances and centimeters for short measurements.
• The imperial system includes units such as inches, feet, and miles for length measurement.
• Converting length measurements often requires understanding the unit you're starting with and the unit you need to end up with.

In practical terms, for our silk measurement, we started with meters, which is a common metric unit and needed to convert it to miles, an imperial unit.

Metric to Imperial Conversion

Switching from metric to imperial units is a common type of conversion. In this exercise, we needed to change the silk length from meters (metric) to miles (imperial).
To perform a metric to imperial conversion:

• First, identify the known value in metric units; here, it was 4,000 meters.
• Use the exact conversion factor, like 1 mile = 1,609.34 meters, to set up an equation that helps convert between systems.
• Divide the metric measurement by the appropriate conversion factor to find the length in imperial units.

For our silk length, dividing 4,000 meters by 1,609.34 meters/mile gives us approximately 2.485 miles. Metric to imperial conversions require precision to ensure the correct unit change.