3.9 Related Rates
Related rates are applications in which two or more
related quantities are changing over time. We want
to compute how fast one quantity changes in terms
of the others.
To Set up a Related Rates Problem:
1. Read the problem carefully. If it helps, draw a
diagram. Label the quantities that vary using
variables. Any quantity that does not change
throughout the entire situation is a constant and
should be labeled with its value
2. Write an equation that relates the variables and
constants
3. List, using derivative notation, the rates given
and the rate you are looking for.
Example 1. Car A is traveling west at 50 mi/h and
car B is traveling north at 60 mi/h. Both are headed
for the intersection of two roads. At what rate are
the cars approaching each other when car A is 0.3
mi and car B is 0.4 mi from the intersection?
Example 2: A water tank has the shape of an
inverted circular cone with base radius 2 m and
height 4 m. If water is being pumped into the tank at
a rate of 2 m3/min, find the rate at which the water
level is rising when the water is 3 m deep.
3.9 Related Rates Worksheet
For each problem,
 Draw a diagram of the situation. Label the quantities that are changing over time
by using variables. Label any constant quantity with its value. Write an equation
that relates the various quantities of the problem
 List the given information and rate(s) and the desired rate. Use derivative
notation to express these rates.
 DO NOT SOLVE THE PROBLEM.
1. At noon, ship A is 150 km west of ship B. Ship A is sailing east at 35 km/hr. and
ship B is sailing north at 25 km/hr. How fast is the distance between the ships
changing at 2 pm?
2. A paper cup has the shape of an inverted cone with height 10 cm and base radius of
3 cm. Water is poured into the cup at a rate of 2 cm3/s. How fast is the water level
rising when the water is 5 cm deep?
3. A spotlight on the ground shines on a wall 12 m away. A man 2 meters tall walks from the
spotlight toward the wall at a speed of 1.6 m/sec. How fast is the length of the man’s
shadow on the wall decreasing when he is 4 m from the building? (Hint: Use similar
triangles)