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# 7.1: Dimensional Analysis

• • Contributed by Pamini Thangarajah
• Professor (Mathematics & Computing) at Mount Royal University
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Thinking out loud

Small pizza is 8" and large Pizza is 16". Are we getting the same amount of pizza if we order two small pizzas instead of one large one?

We all have faced a situation when need to be able to change from one unit of measurement to another unit of measurement. In this section, we discuss a method of converting units called dimensional analysis.

### Time

1 day = 24 hours,

1 hour = 60 minutes, and
1 minute = 60 seconds.

### Length

#### Imperial measurement

1 inch

1 foot= 12 inches

1 yard= 3 feet=36 inches

1 mile =5289 feet

#### Metric

millimeter (mm)

centimeter (cm), 1 cm = 10 mm

meter (m) , 1m= 100 cm= 1000 mm

kilometer (km). 1 km= 1000 m

### Weight

Definition

A unit ratio is a fraction that has a value of 1 if both the numerator and the denominator are expressed in the same units.

Example $$\PageIndex{1}$$:

If the speed limit says 90 kilometres per hour, what is the speed in miles per hour?

We can use dimensional analysis to convert this speed to miles per hour. We can also use reasoning to deduce that we need to divide 90 by 1.6.

## Area

Example $$\PageIndex{1}$$:

John and James have decided to pull up their old carpet and buy a new carpet. The room measures 15 feet by 11 feet, so the area is 165 square feet. However, when they go to the carpet store, they find that the prices are in square yards. How many square yards is their floor?

## Temperature

Temperature: In 1714, a German instrument maker named Gabriel Fahrenheit made the first mercury thermometer. He designated the lowest temperature he could create in the laboratory as 0° and the normal temperature of the body as 98°. On his scale, the freezing point of water is 32° and the boiling point of water is 212°.