Question

Starting with the definition 1 in. = 2.54 cm, find the number of (a) kilometers in 1.00 mile and (b) feet in 1.00 km.

Short answer

(a) 1.60934 km in 1 mile; (b) 3280.84 feet in 1 km.

Step-by-step solution

Understand the Conversion Definitions

To solve the problem, we need to understand the relationship between inches, centimeters, miles, kilometers, and feet. The primary given conversion is 1 inch = 2.54 cm. We also need the conversions: 1 mile = 5280 feet, 1 foot = 12 inches, and 1 kilometer = 1000 meters.

Convert Miles to Kilometers

First, convert the distance from miles to inches using the relationships: 1 mile = 5280 feet and 1 foot = 12 inches. This gives us 1 mile = \( 5280 \times 12 \) inches. Then, convert inches to centimeters using 1 inch = 2.54 cm: \( 5280 \times 12 \times 2.54 \). Lastly, convert centimeters to kilometers using 1 kilometer = 100,000 cm: \( \frac{5280 \times 12 \times 2.54}{100000} \).

Calculate Kilometers in One Mile

Calculate the total using the conversion equation from the previous step. Simplifying, we find \[ 5280 \times 12 \times 2.54 = 160934.4 \] cm and \( \frac{160934.4}{100000} = 1.60934 \) km. Thus, there are 1.60934 kilometers in one mile.

Convert Kilometers to Feet

Next, we start by converting the distance from kilometers to centimeters using 1 kilometer = 100,000 cm. Then, convert centimeters to inches using 1 inch = 2.54 cm: \( \frac{100000}{2.54} \). Convert inches to feet using 1 foot = 12 inches: \( \frac{100000}{2.54 imes 12} \).

Calculate Feet in One Kilometer

Calculate the total number from the conversion equations. Simplifying, we find \[ \frac{100000}{2.54} = 39370.1 \text{ inches} \] and \[ \frac{39370.1}{12} = 3280.84 \text{ feet} \]. Therefore, there are approximately 3280.84 feet in one kilometer.

Key concepts

Miles to Kilometers

To convert from miles to kilometers, you need to understand the conversion relationship between these two units of measure. The core idea here is that 1 mile is equivalent to approximately 1.60934 kilometers. This conversion factor allows you to switch between the two units, making it easier to understand distances whether you're travelling abroad or learning about geography.

Here's a step-by-step breakdown of the conversion process:

• Start with the distance in miles. In this example, we have 1 mile.
• Know that 1 mile equals 5280 feet.
• Then, each foot is 12 inches, so multiply 5280 by 12 to get the total inches in a mile.
• Convert these inches to centimeters by using the conversion 1 inch = 2.54 centimeters.
• Finally, convert the total centimeters to kilometers, knowing that there are 100,000 centimeters in a kilometer.

This series of conversions provides the precise number of kilometers in a mile, thanks to multiplying and dividing by the appropriate conversion factors.

Centimeters to Feet

Converting from centimeters to feet involves a few simple steps and an understanding of the basics of measurement units. Centimeters are a part of the metric system while feet belong to the imperial system, commonly used in the United States.

The conversion process is as follows:

• Start by determining how many centimeters you want to convert. For instance, consider you have 100 centimeters.
• Since 1 inch equals 2.54 centimeters, begin by converting centimeters to inches using the factor, where you divide the number of centimeters by 2.54.
• Once you have the measurement in inches, convert to feet. Remember, 1 foot is made up of 12 inches. So, divide your total inches by 12 to get the answer in feet.

For example, if you start with 328 centimeters, you would convert to inches and then to feet, resulting in approximately 10.76 feet. This type of conversion is helpful for practical tasks like home improvement, where measurements might need to be switched between metric and imperial units.

Conversion Factors

Conversion factors are essential for moving between different measurement units. These factors are the bridges in math that help us transform quantities from one unit to another effortlessly.

To successfully convert units, keep these points in mind:

• Know the basic conversion factors, such as:
  • 1 inch = 2.54 centimeters
  • 1 foot = 12 inches
  • 1 mile = 5280 feet
  • 1 kilometer = 1000 meters or 100,000 centimeters
• Apply these conversion factors in sequence to simplify complex conversions, breaking down the process into manageable steps.
• Remember to keep track of your units at each step to avoid confusion, ensuring both your numerator and denominator in calculations relate directly to your target unit.

Understanding and using conversion factors are vital skills not just for mathematics but also for science and everyday life, as they enable you to adapt easily across different systems of measurement.