How many football fields could fit on a 1 mile stretch of land? Here are the relevant conversion factors:
1 football field = 100 yards
1 mile = 5280 ft
1 yard = 3 ft
A.) The conversion factor relating miles to meters is 1 mile = 1610 m. The speed of light is 3.00 x 10^8 m/s. How fast is this in miles per hour?
B.) The conversion factor relating feet to meters is 1ft=0.305m. Keep in mind that when using conversion factors, you want to make sure that like units cancel leaving you with the units you need. You have been told that a certain house is 164 m^2 in area. How much is this in square feet?
The Correct Answer and Explanation is :
Let’s start by answering your first question:
1. How many football fields could fit on a 1-mile stretch of land?
Given:
- 1 football field = 100 yards
- 1 mile = 5280 feet
- 1 yard = 3 feet
Solution:
First, let’s convert the 1-mile stretch of land into yards.
1 mile = 5280 feet
Since 1 yard = 3 feet, we convert 5280 feet to yards:
[
5280 \, \text{ft} \times \frac{1 \, \text{yard}}{3 \, \text{ft}} = 1760 \, \text{yards}
]
Now, we can find out how many football fields, each 100 yards long, fit into 1760 yards:
[
\frac{1760 \, \text{yards}}{100 \, \text{yards/football field}} = 17.6
]
Therefore, 17 full football fields (with some leftover) would fit on a 1-mile stretch of land.
A.) Speed of Light in Miles Per Hour
Given:
- 1 mile = 1610 meters
- Speed of light = ( 3.00 \times 10^8 \, \text{m/s} )
Solution:
We need to convert the speed of light into miles per hour. We start by converting meters per second into miles per second:
[
\frac{3.00 \times 10^8 \, \text{m/s}}{1610 \, \text{m/mile}} = 1.86 \times 10^5 \, \text{miles/second}
]
Now, we convert seconds to hours (3600 seconds in one hour):
[
1.86 \times 10^5 \, \text{miles/second} \times 3600 \, \text{seconds/hour} = 6.68 \times 10^8 \, \text{miles/hour}
]
Thus, the speed of light is approximately 6.68 × 10^8 miles per hour.
B.) Area Conversion: Square Meters to Square Feet
Given:
- 1 foot = 0.305 meters
- 1 square meter = ( (1 \, \text{m})^2 )
- 1 house area = 164 m²
Solution:
To convert 164 m² to square feet, we first need to convert meters to feet, remembering that the area is in square units:
[
1 \, \text{meter} = \frac{1}{0.305} \, \text{feet} = 3.2808 \, \text{feet}
]
Next, we square this conversion factor to convert square meters to square feet:
[
1 \, \text{m}^2 = (3.2808 \, \text{ft})^2 = 10.7639 \, \text{ft}^2
]
Now, to find the area in square feet:
[
164 \, \text{m}^2 \times 10.7639 \, \text{ft}^2/\text{m}^2 = 1767.5 \, \text{ft}^2
]
Therefore, the house area of 164 m² is 1767.5 square feet.
Explanation (300 words):
The process of converting measurements from one unit to another relies heavily on understanding and using conversion factors. A conversion factor is simply a ratio that expresses the relationship between two different units. For example, to convert from feet to yards, the conversion factor is 1 yard = 3 feet, so we use this ratio to cancel out units and ensure we get the desired outcome.
In the first question, we calculated how many football fields would fit on a 1-mile stretch of land. To do this, we first converted the 1-mile stretch into yards because each football field is measured in yards. We then divided the total number of yards by the length of one football field (100 yards) to determine how many full football fields could fit.
For the second question, we were tasked with converting the speed of light into miles per hour. This required using the relationship between meters and miles, converting the speed from meters per second into miles per second, and then multiplying by the number of seconds in an hour (3600) to get miles per hour.
The third question involved converting square meters to square feet. To do this, we used the conversion factor between meters and feet and squared it, because we are dealing with area (which is measured in square units). After converting 1 square meter to square feet, we multiplied by the given area in square meters to find the equivalent area in square feet.
In all of these problems, the key is recognizing the right conversion factors and applying them correctly to get from one unit to another while ensuring that like units cancel out.