RELATED RATES
Nick Bigner
TJ Howard
Thao Vu
 Related rates problems involve finding a rate at
which a quantity changes by relating that quantity to
other quantities whose rates of change are known.
The rate of change is usually with respect to time.
Steps to solving a related rates
problem

Step 1: draw a diagram related to
the problem and label accordingly

Step 2: specify in mathematical for
the rate of change you are looking
for and record all given
information

Step 3: find an equation involving
the variable whose rate of change
is to be found

Step 4: differentiate with respect
to time (t)

Step 5: state the final answer in
correct form, and specifying the
units that you are using
Some real world
applications of related
rates includes
predicting collisions,
rate at which liquid
drains, travel,
satellites orbiting the
earth, increasing the
volume of an object,
mars expeditions,
pretty much anything
Nasa or any physicists
for that matter do, and
much much more!

Example
Tugboat Problem:
 A tugboat moves a ship
up to the dock by pushing
its stern at a rate of
3m/sec. The ship is 200m
long. Its bow remains in
contact with the dock and
its stern remains in
contact with the pier. At
what rate is the bow
moving along the dock
when the stern is 120m
from the dock?

X2 + y2 = z2

2002 – 1202 = x2

X= 160m

2x(dx/dt) + 2y(dy/dt) = 2z(dz/dt)

(160)(dx/dt) + (120)(-3) = (200)(0)

(160)(dx/dt) = 360
Dx/dt = 2.25m/s
Example
Slag Heap Problem:
 Slag left over from the
manufacturing of cement is
being piled outside the cement
plant. The resulting slag heap is a
cone whose elements make an
angle of 40 degrees with the
horizontal. Environmentalists
measure the circumference of
the heap one day, finding it to be
3000ft, and increasing at 7ft/day.
About how fast is the cement
plant generating slag?
WORK
WORK

V=1/3(п)(r2)(h)

C=2(п)(r2)

R=(c)/2(п)

H=(r)tan40°

V=1/3(п)((c)/2(п))3 tan40°
WORK

dV/dt = (c2)(dc/dt)(tan40°)/(8п2)

dV/dt = (c2)(7)(tan40°)/(8п2)

dV/dt = (30002)(7)(tan40°)/(8п2)
Answer
dV/dt = 669,521.22 ft 3 /day
EXAMPLE
Rectangle Problem
The length of a rectangle is increasing at
3ft/min, and the width is decreasing at 2ft/min.
When the length is 50ft and the width is 20ft, is
the area of the rectangle increasing or
decreasing? At what rate?

LxW=A

(20)(50)=1000

X(dy/dt) + Y(dx/dt) =dA/dt

dA/dt = 3(20) + -2(50)
Answer
decreasing at a rate of
dA/dt = 40ft/min
FRQ
Barn Ladder Problem:
A ladder to the loft in a barn is arranged so
that it can be pushed up against the wall when
it is not in use. The top of the ladder slides in a
track on the wall, and the bottom is free to roll
across the floor on wheels. To make the ladder
easier to move, a counterweight is attached to
the top of the ladder by a rope over a pulley.
As the ladder goes away from the wall, the
counterweight goes up and vice versa.
FRQ Questions
A.
B.
C.
The ladder is 20 ft long. Write an equation expressing
the velocity the counterweight moves as a function of
the distance the bottom of the ladder is from the wall
and the velocity the bottom of the ladder moves away
from the wall.
Find the velocity of the counterweight when the bottom
is 4ft from the wall, and is bring pushed toward the wall
at 3 ft/s.
If the ladder is allowed to drop all the way to the floor
with the bottom moving at 2ft/s and the top remains in
contact with the wall, how fast is the counterweight
moving when the top just hits the floor?

202 = x2 + y2

(20)(0) = x(dx/dt) + y (dy/dt)

Dy/dt = -(x/y)(dx/dt)

V = (x)/((202)-(x2))1/2 (dx/dt)
WORK B

v = (4)/(384)1/2 (-3)

V = -.0623 m/s
WORK C

X = 20

Dx/dt = 2

V = undefined
Try Me!
 Bacteria Spreading Problem: Bacteria are growing in a circular colony one bacterium
thick. The bacteria are growing at a constant rate, thus making the area of the colony
increase at a constant rate of 12 mm^2/hr. How fast is r changing when it equals 3 mm.
Answer
dr/dt = 6/πr
 An
airplane is flying 600 mi/hr on a
horizontal path that will take it directly
over an observer. The airplane is 7mi
high. How fast is z changing when x is
10 mi?
Answer
•X= 10
•Dz/dt= 491.539 which
mean that the
distance is
decreasing at
about 492
miles/hour.

Cone of Light problem: A
spotlight shines on the wall,
forming a cone of light in the
air. The light is being moved
closer to the wall, making the
cone’s altitude decrease at 6ft/
min. At the same time, the
light is being refocused,
making the radius increase at
7ft/ min. At the instant when
the altitude is 3 ft and the
radius is 8 ft, is the volume of
the cone increasing or
decreasing? How fast?
Volume
is decreasing
3
at about 50.3 ft /
min.
Answer
Darth Vader’s spaceship is
approaching the origin
along the positive y-axis at
50km/sec. Meanwhile, his
daughter Ella’s spaceship
is moving away from the
origin along the positive xaxis at 80 km/sec. When
Darth is at y = 1200km
and Ella is at x = 500km,
is the distance between
them increasing or
decreasing? At what rate?
Answer
The distance
is decreasing at
about 15.4m/s
Luke and Leia are trapped
inside a trash compactor on
the Death Star. The side
walls are moving apart at
0.1 m/sec, but the end
walls are moving together
at 0.3 m/sec. The volume
of liquid inside the
compactor is 20 cubic
meters, a constant.
ANSWER
The depth is increasing at 0.02m/s

Foerster, Paul. Calculus Concepts and
Applications. Key Curriculum Press, 1998.
Print.
© Nicholas Bigner, TJ Howard, and Thao Vu