Related Rates


Whenever you start a related rates problem (or most word problems) it's a good idea to:

  • Define a coordinate system: For this we can use the usual: it’s positive if it’s moving to the right or up.  This is to make sure we don’t miss a negative sign somewhere.

  • Draw out the system in question:

  • write out all of the variables that are given and the variables you want to find:

  • We want to find the rate of change for the length of the man’s shadow on the building, or l’.

  • Note that w’ is negative, because as he moves closer to the building at 1m/s, w gets smaller

From there you can find certain other values, such as the distance from the man and the light: x = (20-14) meters = 6 meters, the rate of change of the distance from the man and the light = x’ = -w’ = 1m/s (since x is getting bigger as he moves away from the light), that the rate of change for the distance between the light and building = d’ = 0m/s, and that the rate of change for the man's height, h’ = 0 m/s.

  • If you didn’t see these now, its fine, they would pop up later in the problem and you can find them then.

So the whole picture looks like this:

Now we need to actually relate the two rates w’ and l’. This varies from problem to problem, but usually boils down to a known formula (such as volume or surface area). In this case it's similar triangles. NOTE: Make sure you don’t plug in any number for the variables yet, as we will be taking the derivative of the relation.

  • Formula for similar triangles:

  • Both work, just remember to stay consistent (ie: keep both the numerators as the dimensions of the smaller triangles for the first formula)

Write out the relation:

  • At this point you’d notice we need x, so if you didn't solve for it earlier you can now

  • We want l’, so solve for l first:

  • Then take the derivative, this invokes both the product rule and quotient rule, since the numerator is two variables multiplied with each other (product rule), and those two variables are over another variable in the denominator (quotient rule):

    • Product Rule and Quotient Rule:

  • NOTE: Again, this is where you would find out that you’d need some of those other variables such as h’, d’, and x’ if you missed them earlier.

Now plug in the variables: