What if you went to the grocery store and bought 3 gallons of milk? Could you determine how many pints of milk you purchased? Or how about if you bought 16 pints of milk? How many gallons would this be?
Real-world information is given in dimensions, or the units in which the value is measured. For example, the following are all examples of dimensions.
Analyzing dimensions can help you solve problems in travel, astronomy, physics, engineering, forensics, and quality. Solving problems by converting dimensions or canceling dimensions is the focus of this Concept.
Consider the distance formuladistance=rate⋅time. This formula can be rewritten for rate. rate=distancetime. If distance is measured in kilometers, and time is measured in hours, the rate would have the dimensions kilometershours.
You can treat dimensions as variables. Identical units can divide out, or cancel. For example, kilometershour⋅hour→kilometershour⋅hour→kilometers.
Sometimes the units will not divide out. In this case, a conversion factor is needed. A conversion factor is a multiplier for converting between one set of units to another.
The process of using units or dimensions to help solve a problem is calleddimensional analysis. It is very useful in real world applications as shown in the following problems.
Let's use dimensional analysis to complete the following problems:
Convert 35kilometershour to meters.
Since kilometers ≠ meters, you need to convert kilometers to meters to get the answer. You know there are 1,000 meters in a kilometer. Therefore, you will need to multiply the original dimension by this factor.
Earlier, you were told to suppose that you bought 3 gallons of milk at the grocery store. How many pints of milk did you purchase? How many gallons are in 16 pints of milk?
1 gallon is approximately 8 pints. You can use this conversion factor to determine how many pints are in 3 gallons of milk:
Cross-cancel and multiply:
You bought 24 pints of milk.
To determine how many gallons are in 16 pints of milk, use the same conversion factor:
16pints⋅1gallon8pints Cross-cancel and multiply:
There are 2 gallons in 16 pints of milk.
You are traveling in Europe and want to know how fast to drive to maximize fuel efficiency. The optimal driving speed for fuel efficiency is 55 miles per hour. How fast would that be in kilometers per hour?