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Chapter 8: Using Differentiation to Solve Practical Problems

Problematic Relationships: Related Rates

Related rates problems are the Waterloo for many a calculus student. But they’re not
that bad after you get the basic technique down. The best way to learn them is by
working through examples, so get started!

After working each problem, ask yourself whether the answer makes sense. Asking this
question is one of the best things you can do to increase your success in mathematics
and science. And while it’s not always possible to decide whether a math answer is
reasonable, when it’s possible, this inquiry should be a quick, extra step of every prob-
lem you do.

Q. A homeowner decides to paint his home. He
picks up a home improvement book, which
recommends that a ladder should be placed
against a wall such that the distance from
the foot of the ladder to the bottom of the
wall is one third the length of the ladder. Not 18 ft.
being the sharpest tool in the shed, the
homeowner gets mixed up and thinks that h
it’s the distance from the top of the ladder to
the base of the wall that should be a third of b
the ladder’s length. He sets up his 18 foot
ladder accordingly, and — despite this You don’t have to draw the house — the
unstable ladder placement — he manages to basic triangle is enough. But I’ve sketched
climb the ladder and start painting. a fuller picture of this scenario to make
(Perhaps the foot of the ladder is caught on clear what a bonehead this guy is.
a tree root or something.) His luck doesn’t 2. List all given rates and the rate you’re
last long, and the ladder begins to slide rap- asked to figure out. Write these rates as
idly down the wall. One foot before the top derivates with respect to time.
of the ladder hits the ground, it’s falling at a
rate of 20 feet/second. At this moment, how You’re told that the ladder is falling at a
fast is the foot of the ladder moving away rate of 20 ft / sec. Going down is negative,
from the wall? so
dh 20 db ?
A. Roughly 1.11 feet/second. dt = - dt =
(h is for the distance from the top of the
1. Draw a diagram, labeling it with any ladder to the bottom of the wall; b is for
unchanging measurements and assign- the distance from the base of the ladder
ing variables to any changing things. to the wall.)
See the following figure.
3. Write down the formula that connects
the variables in the problem, h and b.
That’s the Pythagorean Theorem, of
course:
2
2
a + b = c 2 , thus
2
2
h + b = 18 2
continued