Q: "Investment" is a kind of plaster consisting of 1 part silica, 1 part plaster and 1 part water. How many grams of each are needed to fill a circular dish 15 cm in diameter to a depth of 5 mm? Investment has a density of 1.5 g/cm3 .
Q: A tap delivers 2 gallons per minute. How long does it take to fill a tub which measures 2 feet by 3 feet by 8 inches?
A:
Notice that I stacked three equations here, each one equal to "? min." But I ran out of room on the first line, so I continued it on the second, just as I would continue a sentence that ran longer than one line. The first three lines are all part of one long equation. The fourth and fifth lines are a second equation, equal to the first, and the sixth line is a third equation, equal to the other two.
Why did I go to the trouble of writing the second equation at all? Because I wanted to show how you can combine two or more unit factors that are identical. For example, (12 in/1 ft) appears twice. I can simplify my equation by writing (12 in/1 ft)2. I just have to remember that everything inside the parentheses is squared, both the numbers and the units. By contrast, in (1 mL/1 cm3), only the cm is cubed, since the "3" is inside the parentheses. It's really not so hard, but you have to pay attention. Get out your calculator and make sure you can do the arithmetic both ways.
Either you're bored at this point because you've seen this before, or you're bewildered because anything mathematical scares you. If you're bored, you've probably skipped to the next chapter by now anyway, so I'll just add a little more for the bewildered guys.
gal/min to 4 gal/min, I know, intuitively, that it would take half as long to fill the tub. When doubling something in the problem would halve the answer, I know that it goes in the bottom, i.e. for this unit factor, upside down. One more clue is the word "by" in the problem. Just as the verb of a sentence translates into an "=" sign, by translates into multiplication.
Q: "Investment" is a kind of plaster consisting of 1 part silica, 1 part plaster and 1 part water. How many grams of each are needed to fill a circular dish 15 cm in diameter to a depth of 5 mm? Investment has a density of 1.5 g/cm3.
A:
The volume of a cylinder is of course πr2h. I divided the diameter in half to get the radius and I converted 5 mm to 0.5 cm in my head. The tricky part is that I don't know how much a part is, so I just put an "X" in there to hold the spot. Fortunately, whenever I have a recipe given in parts of this to parts of that, the actual number represented by X always cancels out. In this case I get an X in the numerator and an X in the denominator, so they cancel out no matter what X happens to be.
One of the little tricks memes use to ensure their own survival is the mnemonic, a little phrase or jingle intended to jog the memory. So if you forget how Unit Factor Analysis works, here's a little mnemonic to help you remember:
That which is above corresponds to that which is below and that which is below corresponds to that which is above in the accomplishment of the miracle of One Thing. And just as all things come from One, so through the mediation of One, all things follow from this One Thing in the same way.
— The Emerald Tablet of Hermes Trismegistos
That is, the numerator of a unit factor is equal to the denominator and vice versa. Everything in the answer follows from a string of "one things," or unit factors. If you're still bewildered, don't give up. I promise you that I will teach you as little as possible. Instead of giving you different methods for working all the complicated problems you'll run into in this book, I'm going to teach you one method, and we'll use it over and over. Think of it as the Swiss Army Knife for numerical problem solving.
Material Safety
So far, this book has talked about sticks, stones, and other materials with which you have some familiarity. The only hazard in completing this project is the risk of banging your head against a wall as you struggle to learn the intricacies of UFA. But as long as I have your attention for a moment, let me introduce you to Material Safety Data Sheets. MSDS's are required by the United States Occupational Health and Safety Administration (OSHA):
Chemical manufacturers and importers shall obtain or develop a material safety data sheet for each hazardous chemical they produce or import. Employers shall have a material safety data sheet in the workplace for each hazardous chemical which they use.[1]
While required only for hazardous chemicals used in the workplace, many retailers provide MSDS's to consumers upon request. MSDS's are also available online for hazardous and non-hazardous chemicals alike. Since many chemicals have more than one name, the reliability of online searches may be improved by using the Chemical Abstracts (CAS) number instead of, or in addition to the name. CAS numbers for the chemicals discussed in this book are given in each Material Safety section. Though MSDS's include more technical information than most consumers require, they're one of the most readily-available sources of information on hazardous materials and so you should familiarize yourself with them.
Since this project uses no materials at all, you'll introduce yourself to MSDS's by finding them for some common items. Look up charcoal (CAS 7440-44-0), silica (CAS 14808-60-7), and sodium chloride (CAS 7647-14-5). By familiarizing yourself with the hazardous properties of relatively safe materials, MSDS's for materials we will meet later won't seem so intimidating. You may request your sheets from a retailer or you may search for them online using the keyword "MSDS" and the CAS number for the chemical in which you are interested. Particularly on the Internet, there is nothing to prevent the posting of bogus information, so you should always consider the source of your
the MSDS and the potential health effects for eye contact, skin contact, inhalation, and ingestion.
Research and Development
So there you are, studying for a test, and you wonder what will be on it.
• Know the meanings of all of the words that are important enough to be included in the index or glossary.
• Know that the alchemical symbol for air looks like the letter A. The symbol for earth looks like an upside down A.
• Memorize the unit factors listed in Table 3-1.
• Know how to recognize unit factors given in a problem.
• Know the rules codified in the sidebar UFA in a Nutshell .
• Know that fecundity, fidelity and longevity are properties of any successful replicator.
• When you can't remember the last time you missed a unit factor problem, you've probably worked enough examples.
Notes
3.3.
There's nothing to make in this project; it's about brains, not brawn. If you're using this book for a class, you'll probably be given a quiz or test. If you're reading it for fun, you'll need to know this stuff for future chapters. Either way, try working these problems for practice.
Q: A tub measures 2 feet by 3 feet by 8 inches. When filled with water, what is its weight, in pounds?
Q: The density of gold is 19 grams per milliliter. What is the weight, in pounds, of 2.0 liters of gold?
Q: A half inch of rain falls on a field 100 yards long and 200 feet wide. How many gallons of water fell on the field?
Q: A recipe calls for 12 ounces of honey to make 2.0 liters of mead. How much honey is needed to make 5.0 gallons of mead?
Q: If you cut down 6 trees from your neighbor's garden, how many oxen do you have to lend her to make up for it?
Q: Try to summarize in 60 seconds the important events of the past year. Think about the major wars, earthquakes, famines, and epidemics. Think about the elections, discoveries, sporting events, and musical hits. Think about the births, weddings, vacations, and funerals. I imagine that you'll be able to do a pretty decent job condensing a year's events into 60 seconds. Now imagine a cable television channel broadcasting such 60-second summaries 24 hours per day, 7 days per week. How long would it take to broadcast the life history of the typical college freshman? How long would it take to broadcast summaries of each of the approximately 227 years since the United States declared its independence? How long would it take for the approximately 3,753 years since Hammurabi published his Code? How long would it take for the approximately 500,000 years since the domestication of fire?
Q: Imagine that you're standing next to your mother, and she next to her father, and so on back through time. Assume an average of 25 years per generation and a distance of 1 yard per person in line. How far would the line stretch back to contemporaries of the American Revolution, of Hammurabi, and of the domestication of fire?
Q: Imagine that you were able to walk down that line, asking each of your ancestors for a one-minute summary of the most important events in their lives. Assuming that you were to spend 40 hours per week in this activity, how long would it take you to get back to contemporaries of the American Revolution, of Hammurabi, and of the domestication of fire?
Q: A tub measures 2 feet by 3 feet by 8 inches. When filled with water, what is its