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Dimensional Analysis

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 cm^{3} |

1 m^{3} |
= | 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 |

Do the following conversions using the unit factor process. Show
your work for each step and cancel out units as you work. See the
examples on the first page for how you should show your work!
Express your answers in scientific notation when appropriate. **Do all work on a separate piece of paper and number each problem clearly.**

- Convert 4,200 ft to furlongs
- Convert 116 cm to ft
- Convert 64 ft to m
- Convert 2,512 yd to mi
- Convert 13.6 stone to kg
- Convert 175 drams to g

- Convert 0.875 kg to oz
- Convert 54 km to ft
- Convert 42 tbsp to quarts
- Convert 0.5 L to pints
- Convert 788,400,000 seconds to years
- Convert 1,250 cm
^{3}to quarts

The following problems ask you to convert from one metric unit from another. Even if you know the proper metric conversion, do the calculation using the units given in the tables above. After you have finished all of the math, answer the questions below.

- Convert 1 km to m
- Convert 1 g to mg
- Convert 1 L to mL
- Convert 1 kg to g

- Convert 1 m to mm
- Convert 1 cm to mm
- Convert 1 m to cm
- Convert 1 m
^{3}to cm^{3}

- The answers to numbers 1 - 5 in this section were all very close to the same number. What was it?
- Using what you may remember about the metric system and the answer to the previous question, write correct (that is, exact) conversions between km and m, g and mg, L and mL, kg and g, and m and mm.
- What are the exact conversions between cm and mm and between m and cm?
- Why do you think there are so many more cubic centimeters (cm
^{3}) in a cubic meter (m^{3}) than cm in a m?

Some traditional units are not given in decimal form when there are whole units and parts of units. For example, weights for babies and produce are often given in this form—6 lbs 12 oz—rather than this one—6.75 lbs. Write the answers to the following conversions using this convention, which is also used for time units. For example, 2.5 hours is 2 hours 30 minutes.

To convert 10 lbs 2.5 oz to a decimal expressed in pounds you convert the 2.5 oz to pounds and add the result to the 10 lbs.

To convert in the other direction works like this:

14.64 lbs can be written using both pounds and ounces by converting just the 0.64 lbs to oz.

1 lb 2.5 oz × --------- = 0.156 lb 16 ozSo the result is 10. 156 lb.

To convert in the other direction works like this:

14.64 lbs can be written using both pounds and ounces by converting just the 0.64 lbs to oz.

16 oz 0.64 lb × --------- = 10.24 oz 1 lbSo the result is 14 lb 10.2 oz.

- Convert 5 lb 2 oz to decimal pounds
- Convert 12 lb 15 oz to decimal pounds
- Convert 7.88 pounds to pounds and ounces
- Convert 19.27 pounds to pounds and ounces

- Convert 2.75 hours to hours and minutes
- Convert 1.32 hours to hours and minutes
- Convert 3 hours 42 minutes to decimal hours
- Convert 7 hours 19 minutes to decimal hours

One reason this is important is that it enables much quicker arithmetic. Try the following problems to see why. Show your work for each step of each calculation. Do not try to do it in your head!

- Add the following weights:

(3 lbs 3 oz) + (2 lbs 14 oz) - What is the total weight of 7 items that weigh 4 lbs 9 oz each?
- How many blocks of classes can fit into 5 hours if blocks are 1 hr 15 minutes long?
- Convert 5.3 kg to pounds and ounces.

- Convert 3 lb 7 oz to kg
- Add the following weights:

(2.8 lbs) + (5.9 lbs) - What is the total weight of 9 items that weigh 4.25 pounds each?
- What is the total amount of time required to run 3 blocks of classes that are 1.625 hours long?

Here are some fun trivia questions.

- Are you older than 1 million (1,000,000 or 1 × 10
^{6}) seconds? - A billion is a number that is 1,000 times a million (1,000,000,000 or 1 × 10
^{9}). How long would you have to live to be older than a billion seconds?

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 10x^{2}. The x variables combine to be written together as x^{2} instead of x · x. Here are some examples with actual units:

AreaDensity5 cm × 2 cm = 10 cm^{2}5 gVolume------ = 1.67 g/cm^{3}1 m × 2 m × 3 m = 6 m^{3}3 cm^{3}

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 ft^{2}to m^{2}42 ft^{2}42 ft·ft 0.3048 m 0.3048 m ------ = --------- x ---------- x ---------- = 3.9 m^{2}1 1 1 ft 1 ft Enter this into your calculator as 42 × 0.3048 × 0.3048 = 3.90192768

Find the volume in mL of 63 g of gold, which has a density of 19.3 g/mL 63 g 1 mL ------ x --------- = 3.26 mL 1 19.3 g Enter this into your calculator as 63 ÷ 19.3 = 3.264248705

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 (ft^{2}) are really (feet) times (feet) because that’s how area is calculated. Similarly, a cubic meter (m^{3}) 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.

- Convert 1 m
^{3}to cm^{3} - Convert 1 ft
^{2}to in^{2} - Convert 1 mi
^{3}to km^{3} - Convert 1,000 mL (or 1 L) to in
^{3}

- Convert 1 mi
^{2}to km^{2} - Convert 1 gal to cm
^{3} - 1 acre is 43,560 ft
^{2}; convert 1 acre to m^{2} - Convert 42,000,000 gal to m
^{3}

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.

- Highway speed limit: 65 mi/hr

Convert to km/hr - Speed of Light 3.00 ×
10
^{8}m/s

Convert to mi/hr - Speed of Voyager 1 Spacecraft: 17.0 km/s

Convert to mi/hr - Speed of Voyager 1 Spacecraft: 17.0
km/s

How many miles does it travel 10 minutes?

- Speed of a fast snail: 0.28 cm/s

Convert to mi/hr - Speed of Sound in Air: 343 m/s

Convert to mi/hr - Speed of Sound in Water: 3,310 mi/hr

Convert to m/s - Speed of the Space Shuttle in orbit: 17580 mi/hr Convert to km/s
- How long does it take the Space Shuttle to travel 500 mi?

Density is often taught using the formula D = m/V. Since students frequently make mistakes using this formula a better way to proceed is to use dimensional analysis, as in the example shown above. Some useful data: the density of aluminum is 2.70 g/mL; the density of lead is 11.3 g/cm^{3}; the density of water is 1 g/mL.

- Find the mass of 52 mL of aluminum
- Find the volume of 117 g of lead

- Find the mass of 1 in
^{3}of lead - Find the volume of 5 oz. of aluminum