6.5: Related Rates
Objective:
To use implicit differentiation to relate the
rates in which 2 things are changing, both
with respect to time.
Air is pumped into a spherical balloon so that its
volume increases at a rate of 100cm3/s. How fast
is the radius increasing when the diameter is
50cm?
Related Rates Guidelines
1. Make a sketch. Label all sides in terms of variables, even if you are
given the actual values of the sides.
2. Make a list of variables. Separate them into variables that are
constant (never change) and variables that are changing (variables
that are a given value only at a certain point in time). Rates
(recognized by “increasing”, “decreasing” , etc.) are derivatives with
respect to time. They can go into either category. Be aware if the
rate is positive or negative (increasing vs. decreasing over time).
3. Find an equation which ties your variables together. If it is an area
problem, you need an area equation. If it is a right triangle, the
Pythagorean formula may work, etc.
4. Plug in any variable that is constant. NEVER plug in a variable that is
changing.
5. Use implicit differentiation to differentiate you equation with
respect to time, t.
6. Plug in all variables and known rates. Solve for the unknown rate.
7. Label your answers in terms of the correct units ( very important),
and be sure you answered the question asked.
Things to remember……
 Words such as “rate” and “speed” are code words for
derivative, where the underlying variable is time, t.
 Be aware of positive vs. negative rates of change. Is
“something” increasing or decreasing over time?
 If “something” is constant or does not change over
time, the rate of change, or derivative, is 0.
Ex. 1: Air is pumped into a spherical balloon so
that its volume increases at a rate of 100cm3/s.
How fast is the radius increasing when the
diameter is 50cm?
Ex. 2: A 10 ft ladder is resting against a vertical wall when
suddenly it starts to slide, with the bottom moving away
from the wall at a rate of 1 ft/s. How fast is the top sliding
down when the base is 6 feet from the wall?
Ex. 3 An inverted circular cone that has a radius of 2m
and a height of 4 m is being filled with Slime at a rate of
2 m3/min. How fast is the height of the Slime increasing
when it is 3m deep?
Ex. 4: A trough of water is 8 meters long and its ends are
in the shape of isosceles triangles whose width is 5m and
height is 2 m. Water is being pumped in at a constant rate
of 6 m3/sec. At what rate is the height of the water
changing when the water has a height of 1.2 m?
Ex. 5: You’re meeting your friend for breakfast at
Percy’s in Kingston. If you’re driving south at 60mph
and your friend is driving east at 50mph, how fast will
you be approaching them when you’re 0.4 miles away
and they’re 0.3 miles away?