Robert Fuson
Chris Adams
Orchestra Gli Armonici, 100908 Concerto
della Madonna dei fiori, 17 W.A.Mozart,
KV618, Ave Verum Corpus
I could not make this sound work.
Formulas You Need to Know
 Volumes:
 Cone: V=(1/3)∏r²h
 Sphere: V=(4/3)∏r³
 Areas:
 Circle: A=∏r²
 Pythagorean Theorem:
 a²+b²=c²
 Properties of Similar Triangles
What is Related Rates?
 In related rates problems, we are given the rate of
change of certain quantities, and are required to find
the rate of change of related quantities.
 So, in layman’s terms, related rates are problems where
something (usually a distance or volume/area) is being
changed over time.
General Steps for Related Rates
1: Identify the shape being changed and sketch it,
labeling the correct parts with given information.
2: Identify the formula that relates the changing
quantities.
3: Differentiate both sides with respect to time. (d_/dt)
4: Substitute the values given to you by the initial
problem and solve for the desired differentiation (be
sure to use the correct units for your solution).



When dealing with circles (as well as spheres
since they are solved very similarly), all you
need to know is the radius (or diameter, from
which you can easily find the radius) and the
rate of change for the area or the rate of change
for the radius, depending on which you are
being told to find.
First, you take the derivative of both sides form
the initial equation.
Then you plug in ‘r’ and ‘dr/dt’ or ‘dA/dt’ and
solve for the remaining rate of change.

A rock is dropped into a lake. Circular ripples
spread over the surface of the water, with the
radius of each circle increasing at the rate of
3/2 feet per second. Find the rate of change of
the area inside the circle formed by a ripple at
the instant the radius is 4 feet.




First step, notice that this
problem is dealing with a
circle. Now we, draw it,
labeling the parts.
It gives us dr/dt=3/2.
It tells us we are looking
for dA/dt when r=4
Obviously, we’ll use the
equation for area of a
circle, A=∏r²
4ft
4ft
Right Triangles
 When dealing with triangles, the Pythagorean




theorem will be used frequently.
First, use it to find the missing side, since you will
almost never be given all three sides of the triangle.
After finding the missing side, take the derivative of
the Pythagorean theorem, then divide everything by
2 to simplify the equations.
Plug in the sides and the rates that you are given or
can infer to be zero.
Solve for the desired rate
Problem #2
 A 50–foot ladder is placed against a large building.
The base of the ladder is resting on an oil spill, and it
slips at the rated of 3 feet per minute. Find the rated
of change of the height of the top of the ladder above
the ground at the instant when the base of the ladder
is 30 feet from the base of the building.
Problem Examination
 This problem is dealing
with the changing sides of a
triangle, which we draw
using given information.
 dy/dt=5
 Since the length of the
ladder cannot change,
dz/dt=0
 First, we will have to use
the Pythagorean theorem
to find x. Then, we
differentiate both sides and
solve to find dx/dt.
50ft
x
30ft
Finding “x”
 Because this is a
3²+4²=5² triangle, we
can assume that x=40 ft.
 However, to refresh, let’s
run-through the old
fashioned way.
30²+x²=50²
900+x²=2500
X²=1600
X=40
50ft
x
30ft
Solving for dx/dt
x²+y²=z²
x(dy/dt)+y(dx/dt)=z(dz/dt
)
30(3)+40(dx/dt)=50(0)
90+40(dx/dt)=0
dx/dt=(-90)/40
dx/dt=-(9/4)
 dx/dt is negative because
the distance between the
top of the ladder and the
ground is shrinking
50ft
30ft
40ft



Cones are easily the most confusing of the related
rates, relying on the formula for volume of a cone,
V=(1/3)∏r²h.
When given a problem for a cone, you will often
receive the radius and height of the container and
either the current radius or the current height of the
contents of the container.
If this is not the case, you will receive the radius and
height of a changing object, such as an icicle, which
are you are usually asked to evaluate whether of not
it is increasing or decreasing in volume, after being
given an increasing rate of change for height, and a
decreasing rate of change for radius, or visa-versa.






When you are given similar triangles,
you have to compare the two
radiuses to the two heights in order
to find the relationship between the
smaller two.
While you may have the number for
either the smaller height or radius,
do not plug it in at this point, for the
important thing is finding the
relationship between r and h.
To solve, set up as a proportion,
5/r=14/h, then cross multiply.
14r=5h
From here, solve for either r or h,
usually r since most of the time in
these problems, you are asked to
find dh/dt.
r=(5/14)h or h=(14/5)r
http://tutorial.math.lamar.edu/Classes/CalcI/RelatedRates_files/im
age003.gif


After knowing the radius and height of the
cone you’re solving for, whether it be a cone
inside of a larger cone or the large cone, you
are usually asked to solve for dA/dt, or dh/dt.
At this point, you either need to use your
relationship between r and h from earlier to
eliminate either h or r (usually r), or solve take
the derivative of both sides, using the chain
rule for the right side of the equation. Then,
plug in your values and you will be able to
easily determine your answer using simple
algebra.

A cone-shaped icicle is dripping from the
roof. The radius of the icicle is decreasing
at a .2 centimeter per hour, while the
length is increasing at a rate of .8
centimeter per hour. If the icicle is
currently 4 centimeters in radius and 20
centimeters long, what is the volume
rate of change of the icicle?
Here, we are dealing
with a cone, so we
draw and label the
picture.
 We know that
dh/dt=.8cm/hr and
dr/dt=-.2cm/hr (dr/dt is
negative because it is
decreasing)
 We will use the volume
of a cone formula
V=(1/3)r²h, where we
are looking for dV/dt

4c
m
20c
m
4c
m
20c
m
TRY ME! PROBLEM 1

One car leaves a given point and travels north
at 30 mph. Another car leaves the same point
at the same time and travels west at 40 mph.
at what rate is the distance between the two
cars changing at the instant when the cars
have traveled 2 hours?
Try Me! 1 (work page)
TRY ME! PROBLEM 2

A pulley is on the edge of a dock, 15 feet above
the water level. A rope is being used to pull in a
boat. The rope is attached to the boat at water
level. The rope is being pulled in at the rate of 1
foot per second. Find the rate at which the boat
is approaching the dock at the instant the boat
is 20 ft from the dock.
Try Me! 2 (work page)
TRY ME! PROBLEM 3

A spherical snowball is placed in the sun. The
sun melts the snowball so that its radius
decreases ¼ inch per hour. Find the rate of
change of the volume with respect to time at
the instant the radius is 4 inches.
Try Me! 3 (work page)
1971 AB 2

Let y = 2ecos(x)
 Calculate
and
 If x and y both vary with time in such a way
that y increases at a steady rate of 5 units
per second, at what rate is x changing when
Part 1
Part 1 Explanation
For the first derivative you derive it like
you would for any exponential function.
 For the second derivative you use the
Product Rule with a combination of an
exponential function.

Part 2
Part 2 Explanation
Use the first derivative that was found in
part 1 but with a respect to time.
 Then plug what was given in the second
part of the FRQ into the derivative.
 Finally solve for the rate of change of x.

http://people.hofstra.edu/stefan_waner/r
ealworld/tutorials/frames4_4.html
 http://tutorial.math.lamar.edu/Classes/Ca
lcI/RelatedRates.aspx
 Lial, Greenwell, Ritchey, First. Calculus
With Application. Seventh. 2002. Print.

©Chris Adams, Robert Fuson
Bibliography