Problem 56 - Prealgebra

Question

Set up the following conversions as you have been doing. Then perform the calculations on a calculator. The tip of the TV tower at the top of the Empire State Building in New York City is \(1,472\) feet above the ground. Express this height in miles to the nearest hundredth.

Step-by-Step Solution

Answer

The height is approximately 0.28 miles.

Step 1: Understand Feet to Miles Conversion

To convert feet to miles, we need to know the conversion factor. There are 5,280 feet in one mile. This means that to convert from feet to miles, we need to divide the number of feet by 5,280.

Step 2: Set Up the Conversion Equation

Using the conversion factor, we can set up the equation. We know the height of the TV tower is 1,472 feet, so we express this as a fraction of a mile: \[ \text{miles} = \frac{1,472}{5,280} \]

Step 3: Perform the Calculation

Now, we'll perform the division to convert 1,472 feet to miles using a calculator. Enter \(1,472 \div 5,280\) into the calculator. This will give us the result in miles.

Step 4: Round to the Nearest Hundredth

After calculating, we get approximately 0.2787879 miles. To express this to the nearest hundredth, we round to two decimal places. The third decimal place is 8, so we round up: \(0.28\) becomes \(0.28\) as it stays since we're rounding to hundredths.

Key Concepts

Conversion Factor

A conversion factor is like a translation guide that helps you change one unit into another. It's a number that tells you how many of one unit is equal to another unit.
For example, when converting feet to miles, the conversion factor is 5,280 because there are 5,280 feet in a mile.
This means if you have a measurement in feet, you divide it by 5,280 to convert it into miles.

• The conversion factor for feet to miles: 1 mile = 5,280 feet.
• To convert: divide the number of feet by 5,280.

Understanding conversion factors simplifies converting measurements between units. It's like following a recipe, where you know exactly how much of each ingredient you need to achieve the right final product.

Feet to Miles Conversion

Converting feet to miles is straightforward once you know the right steps. Since 5,280 feet make up one mile, this is your key piece of information.
To convert feet to miles:

• First, take your measurement in feet. Let's say it's 1,472 feet, the height of the TV tower at the Empire State Building.
• Next, use the conversion factor for feet to miles by dividing the number of feet by 5,280. Mathematically, it's set up like this:
  \[ \text{miles} = \frac{\text{feet}}{5,280} \]
  Using the tower example: \[ \text{miles} = \frac{1,472}{5,280} \]
• Finally, perform the division to complete the conversion.

By breaking it down into these steps, you can confidently convert any number of feet into miles.

Rounding Decimals

Rounding decimals is a handy way to simplify numbers for easier understanding and use. In this case, once you convert feet into miles, you'll often get a number with several decimal places.
For example, the TV tower height of 1,472 feet converts to approximately 0.2787879 miles. However, such a lengthy decimal isn't always practical.
So, let's make it simpler by rounding to the nearest hundredth (two decimal places):

• Look at the third decimal place (it's 8 in this case, making our evaluation simple.). If it's 5 or more, round the second decimal place up by one.
  
• Here, since we have 0.278, we analyze the third decimal place 8 and decide our rounded value should be 0.28.

Rounding ensures your final answer is easy to read and understand without losing too much accuracy in the context of real-world applications.