Related Rates
M 144 Calculus I
V. J. Motto
The Related Rate Idea

A "related rates" problem is a problem which
involves at least two changing quantities which
change with respect to time.

These problems ask you to figure out the rate at
which one of the quantities is changing given
sufficient information on all the other quantities.

Very often we must bring other information to our
problem to help us.
The Leaning Ladder Problem
The “leaning ladder problem” is a
classical problem in related rates.
Many related rates problems require
that we bring some knowledge to the
problem.
Consider the situation shown at the
right. We can use the Pythagorean
theorem to arrive at an equation that
relates the information. Hence, we
have
x2 +y2 = 102
Classical Ladder Problem
We can apply differential
calculus to our problem if we
assume that x = f(t) and y =
g(t); that is both x and y are
functions of time.
When we differentiate the
constraint equation, we will
need to use the “chain rule.”
This is shown at the right for
you.
x  y  100
2
2
Dt [ x  y ]  Dt [100]
2
2
dx
dy
2x  2 y
0
dt
dy
dx
dy
x y
0
dt
dy
An Illustration - Falling Ladder
Here is an illustration of this situation.
Steps for solving a Related Rate Problem

Steps (continued)
5.
6.
7.
Take the derivative of the equation that
relates the variables with respect to "t"
(remember to make use of the
derivative rules) before you plug the
given information in.
Substitute the given rates and values
into the derivative equation
Solve for the desired quantity
Example 1: The Falling Ladder
A ladder 10 feet long leans against a building. If the
bottom of the ladder slides away from the building
horizontally at a rate of 4 ft/sec, how fast is the ladder
sliding down the house when the bottom of the ladder is
8 feet from the wall.
Example 1 (continued
1.
Draw a picture: Label what you know and do not
know in the diagram
Example 1 (continued)

Example 1: (continued)
5.
Take the derivative of the equation that relates the variables with respect
to "t" (remember to make use of the derivative rules) before you plug the
given information in.
Example 1: (continued)
6.
Substitute the given rates
and values into the
derivative equation
7.
Solve for the desired
quantity.
Example 2: Walking Problem
Two students A and B are walking on
straight roads that meets at right angles.
Student A approaches that intersection at
1 meter per second and student B moves
away from the intersection at 2 meters per
second. At what rate is the angle at
student B changing when A is 10 meters
from the intersection and B is 20 meters
from the intersection.
Example 2 (continued)
Example 2: (continued)
y
tan( ) 
x
 d
sec 2 ( ) 
 dt
1




10

5  d 
1




4  dt 
10
 d

 dt
1 4

 
10 5

2 radians

25 sec
Example 2 (continued)
1
 d 
sec ( ) 
 
10
 dt 
5  d 
1

 
4  dt 
10
2
 d

 dt
1 4

 
10 5

2 radians

25 sec
Can you determine the rate of change for the angle we student A is?
Example 3: Snowball Problem
A spherical snowball with an outer layer of
ice melts so that the volume of the
snowball decreases at a rate of 2 cm3/min.
How fast is the radius changing when the
diameter of the snow ball is 10 cm?
Example 3: (continued)
Example 3: (continued)
dV
2 dr
 (4)( )( r )
dt
dt