Problem 54 - Prealgebra

Question

Set up the following conversions as you have been doing. Then perform the calculations on a calculator. The tallest mountain in the United States is Mount McKinley in Alaska. It is \(20,320\) feet tall. Give its height n miles to the nearest tenth.

Step-by-Step Solution

Answer

Mount McKinley is approximately 3.8 miles tall.

Step 1: Identify Conversion Factor

1 mile is equivalent to 5,280 feet. This is the conversion factor that will allow us to convert feet to miles.

Step 2: Set Up the Conversion Equation

To convert 20,320 feet to miles, use the equation \[\text{miles} = \frac{\text{feet}}{5,280}\]Substituting the given height in feet, we have \[\text{miles} = \frac{20,320}{5,280}\]

Step 3: Calculate the Conversion

Use a calculator to perform the division. Calculate \[\frac{20,320}{5,280} \approx 3.849\]This is the height in miles.

Step 4: Round to the Nearest Tenth

Round 3.849 to the nearest tenth. The digit in the tenths place is 8, and the digit in the hundredths place following it is 4, which means we round down. Thus, \[3.8\]is Mount McKinley's height in miles, to the nearest tenth.

Key Concepts

Understanding Conversion Factors

When converting from one unit to another, a critical aspect is using a conversion factor. A conversion factor is a number used to change one set of units to another, by multiplying or dividing. It acts as a bridge between two different units, allowing for an accurate conversion. In the given exercise, we use the conversion factor of 1 mile being equivalent to 5,280 feet. This is standardized, meaning it is a fixed relationship that always holds true. Conversion factors are essential in ensuring precision and consistency when switching between units. If you're dealing with other units, like feet to inches or kilograms to pounds, you'll have different conversion factors suited for those measurements. Whenever you're tasked with unit conversion, your first step should be identifying the correct conversion factor.

Miles to Feet Conversion Method

Converting miles to feet, or vice versa, involves using the appropriate conversion factor. In this exercise, we are converting from feet to miles. Since we know 1 mile equals 5,280 feet, we set up our conversion equation using this relationship. This equation tells us how many miles there are for a given number of feet. The formula is simple: take the number of feet and divide it by the conversion factor of 5,280.

• Identify the total feet: 20,320 feet in this problem.
• Use the conversion factor: 1 mile = 5,280 feet.
• Divide feet by the conversion factor: \( \text{miles} = \frac{20,320}{5,280} \).

After dividing, the result is approximately \(3.849\) miles. Having this formula in mind helps streamline the process, as you're essentially scaling down from a larger unit of measure to a smaller one.

Rounding Numbers: Understanding the Process

Once an initial conversion is completed, you might need to round the result to make it more comprehensible or to meet exercise requirements. Rounding numbers involves approximating a number to a specified decimal place. In this case, we are rounding to the nearest tenth.

• Look at the number immediately following the tenths position. This tells you if you should round up or down.
• If this digit is 5 or greater, you round up. If it is less than 5, you round down.
• For our conversion of \(3.849\) miles, the tenths place is 8, and the digit after that is 4.

Because 4 is less than 5, the correct action is to round down. So, \(3.849\) rounded to the nearest tenth becomes \(3.8\). Rounding not only simplifies numbers but also provides a more easily understandable quantity, especially useful in real-world applications where overly precise numbers may be unnecessary.