I like to use more variables than are needed because I find it easier to create correct equations. I know that I can always turn the algebra crank later without much thought.
r+s=12 is easy and I know that it's correct. I also know that the half perimeters are pi*r and pi*s.
I then look for enough equations to meet my unknowns. That is what's funny about this problem. You don't have enough information to directly solve for the answer before you look at the choices. There are not enough equations for the variables. Even if you use just r and (12-r), you have no equation, unless, that is, you plug in each answer.
I don't like problems like this, because my first reaction is that you don't have enough information. You do, however, if you look at the possible answers.
Also, why is there no variable in the answer? It's just a unique aspect of this particular problem. What if one of the semicircles is replaced by half of a square? You would have something like this:
4r + (12-r)*pi
for the perimeter. the variable does not disappear when the expression is reduced.
You can't trust what you think because problems try very hard to trick your understanding. You just have to follow the facts (equations) and see where they lead you. As I always say, let the math give you the understanding, not the other way around.
Thursday, September 2, 2010
Steve H on setting up problems
re: how many unknowns?