Related Rates
A fun and exciting application of
derivatives
The Study of Change
• Used to work with real life problems where there
is more than one variable such as
– Rain pouring into a pool
• How fast is the height changing compared to the speed the
volume is changing?
– Falling ladder
• How fast is the base moving away from the house compared
to the speed the top of the ladder is falling towards the
ground?
– Distance between two moving objects
• How fast does the distance between the objects change
compared to the speed of each car?
The Ladder Problem
An 8 foot long ladder is leaning against a
wall. The top of the ladder is sliding down
the wall at the rate of 2 feet per
second. How fast is the bottom of the
ladder moving along the ground at the
point in time when the bottom of the ladder
is 4 feet from the wall ?
Animation(Hopefully)
• http://www2.sccfl.edu/lvosbury/images/LadderNS.gif
Example
• Two cars travel on perpendicular roads
towards the intersection of the roads. The
first car starts 100 miles from the
intersection and travels at a constant rate
of 55 mph. The second car starts at the
same time, 250 miles from the intersection
and travels at a constant speed of 60 mph.
How fast it the distance between them
changing 1.5 hours later?
» From Teaching AP Calculus, McMullin
Two Different Solutions
x
•
•
•
•
Let t = time traveled
X = 100 – 55t
Y = 250 -60t
Z(t) = (100  55t ) 2  (250  60t ) 2
y
z
Differentiate
(100  55t ) 2  (250  60t ) 2
dz 2(100 55t )(55)  2(250 60t )(60)

2
2
dt
2 (100 55t )  (250 60t )
dz 2(17.5)(55)  2(160)(60)

dt
2 17.52  1602
dz
 65.62
dt
Method 2—Easier?
• Differentiate at start
with Pythagorean
Thm
dz
dx
z x y
2
2
dy
2z  2x  2 y
dt
dt
dt
dx
dy
2x  2 y
dz
dt
dt

dt
2z
dz 2(17.5)(55)  2(160)(60)

dt
2 17.52  1602
2
Compare Un-Simplified Versions
dz 2(100 55t )(55)  2(250 60t )(60)

dt
2 (100 55t ) 2  (250 60t ) 2
dz 2(17.5)(55)  2(160)(60)

dt
2 17.52  1602
dz
 65.62
dt
dz
dx
dy
 2x  2 y
dt
dt
dt
dx
dy
2x  2 y
dz
dt
dt

dt
2z
dz 2(17.5)(55)  2(160)(60)

dt
2 17.52  1602
2z
What units?
• The distance between the two cars is
changing at a rate of -65.62 miles per hour
• In general, units of the derivative
•
units of f(x)/ units of independent
variable
Simplified Example
• Suppose x and y are both differentiable
functions of t and are related by the equation
y  x 3
2
• Find dy/dt when x =1, given that dx/dt =2 when
x=1
» From Calculus, 8th e, Larson
Solution
dy
dx
 2x
dt
dt
• Use Implicit
Differentiation
• When x = 1 and dx/dt
=2,
dy
 2(1)( 2)  4
dt