Wow thank you so much MarkFl! I'm a visual learner, so your example is a tremendous help. Any advice on how to improve solving word problems?
Absolutely. There is a method for solving word problems that almost invariably works.
Step 1. In
WRITING, identify, and assign a unique letter, to every unknown or variable that is relevant to the problem at hand. (This is greatly simplified if you can sketch a diagram or some other visual aid.)
It is important to make these identifications in writing because it forces you to be specific. "Well, I'll solve for distance. Let's see: how might I do that. Maybe blah blah." That is a waste of time because you have not yet even really figured out what you are looking for.
u = distance from bottom of taller pole to stake.
v = distance from bottom of shorter pole to stake. This is what I must find.
w = distance from top of taller pole to stake.
x = distance from top of shorter pole to stake.
y = length of wire. This is what I must minimize.
Another important reason to make this identification in writing is that it unburdens your memory. You are trying to think and perform mathematical operations so keeping extra things in memory is just a distraction,
PARTICULARLY IF YOU ARE A VISUAL LEARNER. If you write it down you can
see it.
Step 2. in
WRITING, translate, using your letters, the information in the problem into mathematical form. This frequently involves using some general knowledge to supplement what is in the problem. It always involves translating the specific information in the problem into mathematical form. It also frequently means eliminating as many variables as possible.
y = w + x. If you drew a sketch, this is obvious. And since you have written it down you do not need to keep it in memory. (It might give you partial credit on a test too.)
The rest is not quite so obvious and requires a bit of thought. You might go: "The problem is asking me to find v such that y is minimized. I can do that if I can set up a functional relationship between v and y and set the derivative to zero. But all I know so far gives me y in terms of w and x. Hmm is there any way to use the information in the problem to find a relationship between v and either w or x? And what about u? I have too many variables. Let's look at the problem again."
The problem says that the distance between the poles is 50 feet. Aha that means u + v = 50. Write that down.
But that means u = 50 - v so I can stop worrting about u.
The problem says the poles are vertical. That means the triangles in my sketch are right triangles. Is there any general information about right triangles that the problem will assume I know? Sure. The Pythagorean Theorem.
But that means \(\displaystyle 32^2 + v^2 = x^2 \implies x = \sqrt{v^2 + 1024}.\) I can stop worrying about x.
The Pythagorean Theorem works on the other triangle too.
\(\displaystyle u^2 + 87^2 = w^2 \implies (50 - v)^2 + 7569 = w^2 \implies 2500 - 100v + v^2 + 7569 = w^2 \implies w = \sqrt{v^2 - 100v + 10,069}.\)
Now I have got rid of w.
\(\displaystyle y = \sqrt{v^2 + 1024} + \sqrt{v^2 - 100v + 10,069} = \left(v^2 + 1024\right)^{(1/2)} + \left(v^2 - 100v + 10069\right)^{(1/2)}.\)
At this point I no longer have a word problem I have translated into math lingo.
Step 3. Solve the math problem.
