1 Related Rates
So, what’s a related rate word problem? In these
type of word problems two quantities are related
somehow, like area and radius of a circle. The area,
A, and radius, r, of a circle are related through the
formula
A = πr2
As the radius changes in say ms , the area changes
2
in ms . That is, the rate at which the radius changes
effects the rate at which the area changes. The
rates are related, hence, the title. The derivative
comes into play because we are talking about rates
of change and we can think of the tangent slope as
the instantaneous rate of change of a function.
Now, if we differentiate the function, A = πr2,
with respect to r we get
dA
= 2πr
dr
but there is no reference to time here. Where’s the
“per second” part? What we have is a formula that
relates the change in radius to the change in area
but doesn’t relate the rates. To introduce a time
component we differentiate the equation, A = πr2,
with respect to time. In a sense we are considering
radius and area to be functions of time.
To illustrate this, consider the function rewritten
as
A(t) = π [r(t)]2
and differentiate it with respect to time using the
chain rule. That results in
dr(t)
dA(t)
= 2π [r(t)]
dt
dt
Normally we would skip the formality of rewriting the
equation and our end result would be
dr
dA
= 2πr
dt
dt
This is the formula that relates the rate of change of
the area, dA
dt , with the rate of change of the radius,
dr
dA
m2
dt . Notice that the units for dt are sec because the dA
part is area and is measured in metres2 and the dt is
time measured in seconds in this case. The units for
dr
m
would
therefore
be
dt
sec .
Let’s do two examples to illustrate the related rate
idea in use.
Example
Q?
A spherical balloon is loosing air at a
3
rate of 2 cm
min . How fast is the radius of the balloon
shrinking when the radius is 3 m?
A.
The first sentence tells us how the volume
is changing, that is dV
dt = −2. Notice that we make it
negative because the balloon’s volume is shrinking,
the change is negative. The question wants us to
find dr
dt . First we must decide how the volume and the
radius are related. We use the volume of a sphere
formula.
4 3
V = πr
3
Now differentiate that formula with respect to time to
get
4
dV
dr
= π3r2
dt
3
dt
2 dr
= 4πr
dt
Now substitute dV
dt = −2, r = 3 into the equation and
solve for dr
dt .
2 dr
−2 = 4π(3)
dt
−1
dr
=
dt
18π
dr
−1 cm
Therefore dt = 18π min .
Example
Q?
The water in a cylindrical glass is rising at
a rate of 1 cm
sec . The radius of the glass is 2 cm. What
is the rate at which water is being poured into the
glass?
A.
The formula for the volume of a cylinder is
V = πr2h
where h is the height and r is the radius. This is
the formula that relates volume and height. We want
the rate of change of volume, dV
dt . Differentiating
the above formula with respect to time involves the
product rule because both radius and height can be
thought of as functions of time. This yields
dr
dV
2 dh
= 2πr h + πr
dt
dt
dt
Note that we know that the radius is constant so we
could have treated it as such but the math will take
dr
care of it. We now set dh
=
1,
dt
dt = 0 and r = 2 to get
dV
= 2π(2)(0)h + π22(1)
dt
= 4π
3
Therefore the volume is increasing at 4π cm
sec .
So, when I see a word problem, how do I know
it’s a related rate problem? The word rate is usually
3
there, but also the presence of units like ms or cm
min
could signal a related rate question. Then you write
the function that relates the variables in question,
differentiate it with respect to time, substitute for
the known quantities and solve for the one left over
unknown. Remember to include units in your answer.
Also look at the examples in section 4.1 of
the text.
Section 4.1, #1-3, 5-12, 15, 18, 21, 27
Submit # 6, 12, 18 on Monday.
Use the odd number questions to verify that you know
what you are doing.
Example 1 Oil spills from an Exxon tanker in a huge circum
lar slick. The radius of the slick is increasing 10 min
. How
fast is the area of the slick increasing 1 hour after the spill
started?
Example 2 A canoe is being pulled into a dock by a rope
attached to the bow and passing thru a pulley on the dock
that is 1 metre higher than the bow of the canoe. If the
rope is being pulled in at a rate of 1 ms , how fast is the canoe
approaching the dock when it is 8 m from the dock?
Can we create a function that relates speed of the canoe
with the distance from the dock?
Example 3 I was in lock at the canal near Pittsford has a
trapezoidal cross section that is 30 feet at the bottom and
40 feet at the top. The total height is 40 feet. The lock is
ft
60 feet long. If my boat is dropping at 1 min
at the halfway
point, how fast is the water being drained from the lock?
Example 4 Sound intensity various inversely with the square
of the distance from the speaker stacks. Decibels are measured withe the equation
· ¸
I
D = log
I0
What is the rate of change of the decibel levels as I walk
away from the speaker stack at 3 ms when I am 10m away.