These are some systematic
, clear steps that students can take to help in solving a related rates problem:
Step 1, Draw a figure and label the quantities that vary.
In a related rates problem, you'll always have at least two varying rates. Sometimes more. Keeping track of these is difficult, but vital to solving the problem, so don't muck things up from the beginning by trying to figure it out in your head. If a figure is provided, redraw the figure and label what information was provided in the problem's setup. If not, imagine the scene described and draw the best you can. It will at least get you in the right frame of mind to have to consider how things are placed, even if the path seems clear. You may catch something you might've missed. Besides all that, it looks purtty on paper.
Step 2, Identify the rates of change that are known and the rate of change that is to be found.
It sucks to grind through a problem missing a piece of information that you could've known from the beginning. Don't let it happen, keep your bases covered by writing out everything you know for sure. This is important in any word problem, but especially for related rates, because you will not get anywhere if you miss a provided piece of information. They're designed that way (the bastards!). Be careful, though, of extraneous info.
Step 3, Find an equation that relates the quantity whose rate of change is to be found to the quantities whose rates of change are known.
This will be a set-up with static variables, like just plain ol' x, y, or z. One variable may also be something more complicated, like volume, surface area, or angle. Just make sure it's accurate. You need this equation for the next step.
Step 4, Differentiate both sides of this equation with respect to time and solve for the derivative that will give the unknown rate of change.
You're almost homefree, substitute substitue substitute! This is when you take that nice list of known things at the beginning and plug it in like mad. Hopefully (prey hard), you'll just have your unknown rate of change left. It can either be a constant, or an equation in terms of the variable defining the rate of change. If the latter, there's one last step.
Step 5 (optional), Evaluate this derivative at the appropriate point.
Plug, chug, scratch down the answer and you're done. Congrats! Time to move on to the next one... *sigh*
Steps sourced from Anton, Howard. Calculus: A New Horizon, 6th ed. New York: John Wiley & Sons, 1999. Explication my own work