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## Group Activity: Dimensional Analysis Part 3

 Linear Measure 2.54 cm = 1 in 12 in = 1 ft 5280 ft = 1 mi 0.3048 m = 1 ft 1 km = 0.621 mi 1 furlong = 660 ft 3 ft = 1 yd 0.0394 in = 1 mm
 Weight/Mass 1 stone = 14 lbs 1 lb = 0.4536 kg 1 oz = 16 drams 1 oz = 28.3 g 16 oz = 1 lb 2,000 lbs = 1 ton 1 oz = 28,350 mg 1 carat = 0.2 g
 Volumetric Measure 1 cup = 16 tablespoons 2 cups = 1 pint 2 pints = 1 quart 1 gal = 4 quarts 1 gal = 3.78 L 1 mL = 1 cm3 1 m3 = 1000 L 1 cup = 236.6 mL
 Time 1,000 ms = 1 s 60 s = 1 min 60 min = 1 hr 24 hr = 1 day 365 day = 1 yr 14 days = 1 fortnight 10 yr = 1 decade 100 yr = 1 century

#### Combining Units

So far we have concentrated on cancelling units when using dimesional analysis. Units can also be combined according to the rules of algebra. For example, 12x multiplied by 9/4x equals 27 because the x cancels out and 12/4 × 9 is 27. If the problem had been 2x multiplied by 5x then the answer is 10x2. The x variables combine to be written together as x2 instead of x · x. Here are some examples with actual units:

```Area                        Speed
5 cm × 2 cm = 10 cm2         42 mi
Volume                     ------ = 42 mi/hr
1 m × 2 m × 3 m = 6 m3       1 hr
```

The general idea is that if a unit does not cancel out during the calculation, it is still there at the end. Combine like terms and pay attention to whether the unit is in the numerator (on top) or in the denominator (on the bottom).

Here are two examples of problems solved by using combined units.

``` Convert 42 ft2 to m2
42 ft2   42 ft·ft    0.3048 m      0.3048 m
------ = --------- x ---------- x ----------  = 3.9 m2
1         1           1 ft          1 ft
Enter this into your calculator as 42 × 0.3048 × 0.3048 = 3.90192768
```

``` Find the time to travel 50 miles at 35 mi/hr
50 mi      1 hr
------ x ---------   = 1.43 hr or 1 hr 25.7 min
1        35 mi
Enter this into your calculator as 50 ÷ 35 = 1.428571429
```

##### Areas and Volumes

Areas and volumes are examples of combined units, for the most part. An acre is an example of a unit of area which is not a combined unit. A gallon is similarly a unit of volume which is not a combined unit. Square feet (ft2) are really (feet) times (feet) because that’s how area is calculated. Similarly, a cubic meter (m3) can be imagined as a cubic space one meter on a side. For area units you must convert with two conversion factors. For volume, use three conversion factors.

1. Convert 0.25 m3 to cm3
2. Convert 5 ft2 to in2
3. Convert 42 mi3 to km3
4. Convert 1,000 mL (or 1 L) to in3
1. Convert 15 mi2 to km2
2. Convert 10 gal to cm3
3. 1 acre is 43,560 ft2; convert 1 acre to m2
4. Convert 42,000,000 gal to m3

##### Speeds

Common units of speed are miles per hour (mi/hr), kilometers per hour (km/hr), meters per second (m/s), centimeters per second (cm/s), kilometers per second (km/s), and miles per second (mi/s). Show work as always. There is an example of this type of calculation in the introduction.

1. Highway speed limit: 65 mi/hr
Convert to km/hr
2. Speed of Light 3.00 × 108 m/s
Convert to mi/hr
3. Speed of Voyager 1 Spacecraft: 17.0 km/s
Convert to mi/hr
4. Speed of Voyager 1 Spacecraft: 17.0 km/s
How many miles does it travel 10 minutes?
5. Speed of Voyager 1 Spacecraft: 17.0 km/s
How long does it take to travel 1,000,000 mi?
1. Speed of a fast snail: 0.28 cm/s
Convert to mi/hr
2. Speed of Sound in Air: 343 m/s
Convert to mi/hr
3. Speed of Sound in Water: 3,310 mi/hr
Convert to m/s
4. Speed of the Space Shuttle in orbit: 17580 mi/hr
Convert to km/s
5. How long does it take the Space Shuttle to travel 3,000 mi? (This is the approximate width of N. America)

Dimensional Analysis Part 1
Dimensional Analysis Part 2
Metrics Units are treated in a separate activity.