Question

The distance from the center of the Moon to the center of the Earth ranges from approximately \(356,000 \mathrm{~km}\) to \(407,000 \mathrm{~km}\). What are these distances in miles? Be certain to round your answers to the appropriate number of significant figures.

Short answer

Answer: The range of distances is approximately 221,000 miles to 253,000 miles.

Step-by-step solution

Identify the conversion factor

First, we need to find the conversion factor for kilometers to miles. The appropriate factor is 1 kilometer is equal to 0.621371 miles.

Convert the minimum distance from kilometers to miles

To convert the minimum distance (356,000 km) from kilometers to miles, we use the conversion factor that we have found in the previous step: Minimum distance in miles = (356,000 km) * (0.621371 miles / 1 km) Minimum distance in miles ≈ 221288.196 miles

Round the minimum distance to the appropriate number of significant figures

We were given the distance in kilometers as 356,000 km, which has 3 significant figures. So, we need to round our answer to 3 significant figures as well: Minimum distance in miles ≈ 221,000 miles (rounded to 3 significant figures)

Convert the maximum distance from kilometers to miles

Now, we'll do the same for the maximum distance (407,000 km): Maximum distance in miles = (407,000 km) * (0.621371 miles / 1 km) Maximum distance in miles ≈ 252977.677 miles

Round the maximum distance to the appropriate number of significant figures

Again, we were given the distance in kilometers as 407,000 km, so we need to round our answer to 3 significant figures: Maximum distance in miles ≈ 253,000 miles (rounded to 3 significant figures)

Final answer

So, the distances from the center of the Moon to the center of the Earth range from approximately 221,000 miles to 253,000 miles, rounded to the appropriate number of significant figures.

Key concepts

Significant Figures

Significant figures are crucial in scientific calculations because they communicate the precision of measurements. When a value is expressed in significant figures, it indicates which digits are meaningful - based on the accuracy of the instruments used in acquiring that measurement.
For example, if we have a distance measured at 356,000 km, it includes 3 significant figures: 3, 5, and 6. This means that this distance has a precision up to the ten-thousands place.

• Leading zeros are not significant
• Trailing zeros are only significant if they follow a decimal point
• All non-zero numbers are always significant

When converting units, like kilometers to miles, ensuring that the precision of the results matches the precision of the input values is important. Therefore, the distances should be rounded to the same number of significant figures they were measured in: 356,000 km to 221,000 miles (each with 3 significant figures).

Distance Measurement

Distance measurement is the determination of the physical space between two points. Different tools and units are used based on the scale of measurement and precision required.

• In everyday life, tools like rulers, measuring tapes, and odometers are used.
• In scientific contexts, laser range finders, GPS, and other sophisticated sensors may be involved.
• Measurement units vary too, from centimeters and meters to kilometers for larger distances.

In astronomy, effectively measuring the distance between celestial bodies like Earth and the Moon is vital, as it affects various scientific calculations and understandings. The distances are often large and require accurate conversion between units to be meaningful in different contexts, such as interchanging kilometers and miles. This conversion helps bridge the gap between scientific data and everyday understanding.

Kilometers to Miles Conversion

Converting kilometers to miles is a common task, especially to make scientific data accessible to an audience familiar with imperial units. The conversion factor between kilometers and miles is essential for accurate conversion.
By using the readily available conversion factor: 1 km equals approximately 0.621371 miles, we transform the distance from kilometers to miles.

• Multiply the distance in kilometers by the conversion factor: \[ \text{Distance in miles} = \text{Distance in kilometers} \times 0.621371 \]
• Round the result to the appropriate number of significant figures to align with the precision of the original measurement.

As demonstrated in the solution, the Moon's distance from Earth, ranging from 356,000 km to 407,000 km, converts to 221,000 miles and 253,000 miles, respectively, both rounded to 3 significant figures. This process ensures a clear and precise communication of these sizeable outer space distances.