- The variables names I chose were not pulled out of a hat. I deliberately
picked names that correspond in some nice/‘obvious’ way to what it
represents. For example, notice that all the ‘Cost’ based variables are , and
moreover, that ‘
something’ clues you into what the variable is the cost of; for patio, for labor, for (paver) brick. Thus by intelligently naming your various, you can often make things more sensible.
- We have generalized everything we possibly could and as I mentioned earlier, this is almost always overkill. For now it’s helpful to see what the possibilities are for generalizing, then we will want to cut back to which specific data would be helpful to generalize.
- Despite the first point above, the variable names may make sense in context, but it will be easy to forget that context if we were to put this down and come back to it in six months. For this reason, it’s always a good idea to explicitly write down all your variables and what they literally mean. This means writing something like ” = area of the patio in square feet” not = area of patio. Units are the easiest thing to forget, and typically the leastsomewhat plausiblemost likely area to cause errors... just ask NASA! .
This section explains types and interactions between variables.
To best understand variables and their roles, let’s revisit the patio situation and consider each of the pieces of data as “potential generalized data”. In our patio example we might consider the following pieces of data;
- The length and width of the (rectangular) patio.
- The area of the patio.
- The cost of the cement pavers.
- The size (dimensions; length and width) of a single cement paver.
- The number of necessary cement pavers to make the patio.
- The cost of labor to build the patio.
- The time to build the patio.
Thus, we could write down the following relationships:
- The (Number of Pavers Needed) is approximately the (Area of the Patio) divided by the (Area of Pavers).
- The (Area of Pavers) is equal to the product of the (Width of Pavers) and the (Length of Pavers).
- The (Area of Patio) is the product of the (Width of Patio) and the (Length of Patio).
- The (Total Cost of Pavers) is the product of (Cost of a Paver) and (Number of Pavers Needed)
- The (Total Cost of Labor) is the product of the (Hourly Cost of Labor) and the (Time to Build Patio (in hours)).
- The (Total Cost of Patio) is the sum of the (Total Cost of Pavers) and the (Total Cost of Labor).
Look at the list above and try to read them out loud. Really; go do it. How far did you get before you stopped to come back and read this because it was obnoxious to read all those words for “such obvious relationships”? It should be clear that, although we have to write phrases like “Area of Pavers” and “Width of Patio”, it would be a lot easier if we could encode this information in something faster and easier to read; after all, we know what we mean right?This is where variables come into play. We could build an encoding, a kind of “quick reference sheet” for a shorthand to refer to these things. An example of such a thing might be the following;
- is the number of bricks needed to build our patio.
- is the area of the patio (in square feet).
- is the area of a paver brick.
- and are the length and width of the patio respectively.
- and are the length and the width of the paver bricks respectively.
- is the cost of the labor (in dollars).
- is the cost of the paver bricks (in dollars).
- is the total cost of the patio project (in dollars).
- is the total time to build the patio.
The above looks intimidating. That’s an awful lot of letters, but there are a few things to keep in mind when looking at that list.
Let’s say you work for a construction company and you are asked to make a model to determine the price of building a patio for a customer. That customer has very specific dimensions that they want, and so you know you will be given the length and width of the patio, but your calculations require area. In this case you would use the original equation above; Here the relationship between these variables in your model expects to have you supply length and widtharea and lengtharea and width of the patio (the given information from your customer) and it will in turn calculate the areawidthlength of the patio.
Now let’s say a customer is planning an above-ground pool. They know they need a certain amount of minimum square footage to have a foundation for their pool, a grill, and a lounge area. Furthermore they know they want the patio to run the full width of their main porch area. So when building the model, you want to build it assuming you will know the area and the width, but not the length. Thus you would use the following relationship;
In both these cases we have the same variables and the model has the same end goal (to calculate the cost to build a patio), but in one situation we expect to know the length and width, but need the area. In the other situation you expect to know the area and width, but need the length.
There are other kinds of variables one could encounter as well; of specific importance in calculus is the arbitrary constant. This is typically a result of some initial information used in your model, and is a byproduct of choices in your model, but they are unaffected by independent variables. One can think of the arbitrary constant as being a sort of “starting spot” for your model. That is to say, even though your “starting spot” is typically of great importance to your outcome (your starting height when throwing a ball and measuring how far it goes for example) no matter what information you “feed into” the model (eg throwing speed, throwing angle, etc), your starting spot doesn’t change. Thus the arbitrary constant doesn’t change based on any of these “input” values.
One might wonder what the difference is, then, between a constant and an arbitrary constant. It may be clear that it doesn’t vary based on independent variables, so it seems like it is constant. The key point though is that you may not know the intended “starting spot” (ie the height someone will throw from) of your model when you are designing it.
Consider our patio example again. You are building a generic model to calculate the cost of building a patio for your company, and part of that is travel costs. The travel costs themselves will depend on the location (where the customer lives, how accessible the construction site is, etc). But once you have determined the cost for travel, it won’t change depending on the size of the patio (the independent variable). Thus it will be a constant value, but one that depends on the customer’s location, not the customer’s specific project. This is cost would then be an arbitrary constant; something that varies from project to project (specific model to specific model) but is constant within the specific model.