3.5. RELATED RATES
201
From this last equation, differentiating with respect to t implies
dV
dt
=
1
2
πr
2 dr
dt
,
from which the same conclusions made earlier about
dr
dt and
dh
dt can be made.
Our work with the sandpile problem above is similar in many ways to our approach
in Preview Activity 3.5, and these steps are typical of most related rates problems. In
certain ways, they also resemble work we do in applied optimization problems, and here
we summarize the main approach for consideration in subsequent problems.
• Identify the quantities in the problem that are changing and choose clearly defined
variable names for them. Draw one or more figures that clearly represent the
situation.
• Determine all rates of change that are known or given and identify the rate(s) of
change to be found.
• Find an equation that relates the variables whose rates of change are known to those
variables whose rates of change are to be found.
• Differentiate implicitly with respect to t to relate the rates of change of the involved
quantities.
• Evaluate the derivatives and variables at the information relevant to the instant at
which a certain rate of change is sought. Use proper notation to identify when a
derivative is being evaluated at a particular instant, such as
dr
dt
r=4 .
In the first step of identifying changing quantities and drawing a picture, it is important
to think about the dynamic ways in which the involved quantities change. Sometimes a
sequence of pictures can be helpful; for some already-drawn pictures that can be easily
modified as applets built in Geogebra, see the following links 2 which represent
• how a circular oil slick’s area grows as its radius increases http://gvsu.edu/s/9n;
• how the location of the base of a ladder and its height along a wall change as the
ladder slides http://gvsu.edu/s/9o;
• how the water level changes in a conical tank as it fills with water at a constant rate
http://gvsu.edu/s/9p (compare the problem in Activity 3.14);
• how a skateboarder’s shadow changes as he moves past a lamppost
http://gvsu.edu/s/9q.
2 We again refer to the work of Prof. Marc Renault of Shippensburg University, found at
http://gvsu.edu/s/5p.
201
From this last equation, differentiating with respect to t implies
dV
dt
=
1
2
πr
2 dr
dt
,
from which the same conclusions made earlier about
dr
dt and
dh
dt can be made.
Our work with the sandpile problem above is similar in many ways to our approach
in Preview Activity 3.5, and these steps are typical of most related rates problems. In
certain ways, they also resemble work we do in applied optimization problems, and here
we summarize the main approach for consideration in subsequent problems.
• Identify the quantities in the problem that are changing and choose clearly defined
variable names for them. Draw one or more figures that clearly represent the
situation.
• Determine all rates of change that are known or given and identify the rate(s) of
change to be found.
• Find an equation that relates the variables whose rates of change are known to those
variables whose rates of change are to be found.
• Differentiate implicitly with respect to t to relate the rates of change of the involved
quantities.
• Evaluate the derivatives and variables at the information relevant to the instant at
which a certain rate of change is sought. Use proper notation to identify when a
derivative is being evaluated at a particular instant, such as
dr
dt
r=4 .
In the first step of identifying changing quantities and drawing a picture, it is important
to think about the dynamic ways in which the involved quantities change. Sometimes a
sequence of pictures can be helpful; for some already-drawn pictures that can be easily
modified as applets built in Geogebra, see the following links 2 which represent
• how a circular oil slick’s area grows as its radius increases http://gvsu.edu/s/9n;
• how the location of the base of a ladder and its height along a wall change as the
ladder slides http://gvsu.edu/s/9o;
• how the water level changes in a conical tank as it fills with water at a constant rate
http://gvsu.edu/s/9p (compare the problem in Activity 3.14);
• how a skateboarder’s shadow changes as he moves past a lamppost
http://gvsu.edu/s/9q.
2 We again refer to the work of Prof. Marc Renault of Shippensburg University, found at
http://gvsu.edu/s/5p.