Seeming paradox when squaring distance depending on units

In summary, the conversation discusses the incorrect use of conversion factors when converting feet to miles squared. The correct conversion factor for this conversion is 27878400 feet squared equals 1 mile squared.
  • #1
CuriousBanker
190
24
Hello all

This is probably simple and I'm overlooking something

1 mile = 5280 feet

10% of a mile is 528 feet

528 feet squared is 278,784 feet which is 52.8 miles squared

But 0.1 miles squared is .01 miles squared

So depending on if you square it as 0.1 miles, or if you convert it to feet, then square it and convert it back, you get wildly different answers

Why is this.? Not sure why I thought of this.

Thanks in advance
 
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  • #2
528 feet squared is 278,784 feet which is 52.8 miles squared
You used the wrong conversion factor.

First of all, (528 feet)2 is not equal to 278784 feet. It is equal to 278784 feet2.

In any case, you want to convert this to a value in miles2.

Now, the conversion factor between feet2 and miles2 is not the same as the conversion factor between feet and miles, because feet2 are not the same as feet, and miles2 are not the same as miles. In fact, the conversion factor is 27878400 feet2 = 1 mile2, and if you use this correct conversion factor you will get the correct answer of 0.01 miles2.
 
  • #3
CuriousBanker said:
528 feet squared is 278,784 feet

That should be feet squared.

##\frac{278784 ft^2}{1} \cdot \frac{(1 mi)^2}{(5280ft)^2} = 0.01 mi^2##
 

FAQ: Seeming paradox when squaring distance depending on units

1. How can the distance between two points seem to change when squared, depending on the units used?

When we square a distance measurement, we are essentially multiplying the distance by itself. Since different units represent different amounts, this can result in a seeming paradox. For example, if we have a distance of 2 meters and square it, we get 4 square meters. However, if we have a distance of 200 centimeters and square it, we get 40,000 square centimeters. Even though the distance between the two points is the same, the squared result appears different due to the units used.

2. Why is it important to consider units when dealing with squared distance?

Units are crucial in understanding and communicating measurements. When we square a distance, we are essentially finding the area of a square with sides equal to that distance. Therefore, the resulting units will be squared as well. If we do not consider the units, it can lead to confusion and incorrect calculations.

3. How does this seeming paradox relate to the concept of dimensional analysis?

Dimensional analysis is a method used to convert between units. It helps us understand the relationship between different units and how they can be converted into each other. The seeming paradox when squaring distance is a perfect example of why dimensional analysis is important. It shows how units can affect our perception of measurements and how they can be converted to accurately represent the same distance.

4. Can this paradox occur with other types of measurements?

Yes, this paradox can occur with any type of measurement that involves squaring or cubing. For instance, if we have a volume of 2 cubic meters and convert it to cubic centimeters, the result will be 2,000,000 cubic centimeters. Even though the volume is the same, the units used make it seem different. This is why it is important to be aware of units and how they affect measurements.

5. How can we avoid confusion when dealing with squared distance and units?

To avoid confusion, it is important to always keep track of the units used and convert them as needed. It can also be helpful to use visual aids, such as diagrams or graphs, to better understand the concept. Additionally, double-checking calculations and consulting with others can help ensure accuracy and prevent misunderstandings caused by units and squared measurements.

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