Section 4.6
Related Rates
• We have seen that if we can model a situation
by a continuous function, we can use calculus
to analyze changes that have taken place
• We have been concerned with how the rate of
change in the output variable is affected by a
change in the input variable
• Now we will look at the interaction of the rates
of change of input and output variable with
respect to a third variable
• The interconnection of the input and output
variables is reflected in an interaction between
their rates with respect to a third variable
• Consider the following problem:
– A spherical balloon of radius r centimeters has a volume
given by V  (4 / 3)r 3
• Find dV/dr when r = 1 , r = 2 and r = 3 and give a practical
interpretation of your answers.
• Suppose that the balloon is being inflated in such a way that r(t)
= 2t centimeters after t seconds. How fast is the volume of the
balloon increasing when r = 1? When r = 2?
• Air is being blown in the balloon at a constant rate of 50 cubic
centimeters per second. How fast is the radius of the balloon
increasing when r = 1? When r = 2?
– In order to answer the last two questions we need a
technique called implicit differentiation
Implicit Differentiation
• Implicit differentiation involves differentiating
both sides with respect to a third variable and
then solving for the derivative
• Take our function
V  (4 / 3)r 3
• We will take the derivative of both sides with
respect to t (time) using the chain rule
• Then we can look at finding how the radius is
changing
• A spherical balloon of radius r centimeters has a
volume given by V  (4 / 3)r 3
– Suppose that the balloon is being inflated in such a way
that r(t) = 2t centimeters after t seconds. How fast is the
volume of the balloon increasing when r = 1? When r =
2?
– Air is being blown in the balloon at a constant rate of 50
cubic centimeters per second. How fast is the radius of
the balloon increasing when r = 1? When r = 2?
5 Steps to solving related rates problems
1. Read problem and identify variables involved.
Identify independent and all dependent
variables.
2. Use the given equation (or find one) relating the
variables.
3. Differentiate both sides of the equation with
respect to the independent variable.
4. Substitute in the known quantities and rates into
the related rates equation and solve for the
unknown rate
5. Interpret in the context of the problem
A ladder 10 feet long rests against a vertical
wall. If the bottom of the ladder slides away
from the wall at a rate of 1 ft/s, how fast is the
top of the ladder sliding down the wall when
the bottom of the ladder is 6 ft from the wall?
A water tank has the shape of cylinder with
base radius 2m and height 4m. If water is
being pumped into the tank at a rate of
2m3/min, find the rate at which the water level
is rising when the water is 3m deep.
A water tank has the shape of an inverted
circular cone with base radius 2m and height
4m. If water is being pumped into the tank at a
rate of 2m3/min, find the rate at which the
water level is rising when the water is 3m
deep.