Related Rates
An application of the derivative
Formulas you should know
Trust me!
Formulas
Circles
Triangles :
A  r
Pythagorean Theorem
2
C  2 r
a b  c
2
2
Area
Rectangular Prisms
v  lwh
SA  2lw  2lh  2wh
1
A  bh
2
2
Formulas
Cylinders
Spheres
V  r h
LSA  2 rh
4 3
V  r
3
2
SA  4 r
2
SA  2 rh  2 r
2
Right Circular Cone
V
r h
2
3
LSA   r r  h
2
2
SA   r r  h   r
2
2
2
If one of these is on the test,
I’ll give you the formula!
Example #1
 A circular pool of water is expanding at
the rate of 16π in2/sec. At what rate is
the radius expanding when the radius
is 4 inches?
Example #2
 A 25-foot ladder is leaning against a
wall and sliding toward the floor. If the
foot of the ladder is sliding away from
the base of the wall at a rate of 15
ft/sec, how fast is the top of the ladder
sliding down the wall when the top of
the ladder is 7 feet from the ground?
Example #3
 A spherical balloon is expanding at a
rate of 60π in3/sec. How fast is the
surface area of the balloon expanding
when the radius of the balloon is 4
inches?
Example #4
 An underground conical tank, standing
on its vertex, is being filled with water
at a rate of 18π ft3/min. If the tank has
a height of 30 feet and a radius of 15
feet, how fast is the water level rising
when the water is 12 feet deep?
Example #5
 A circle is increasing in area at a rate of
16π in2/sec. How fast is the radius
increasing when the radius is 2 inches?
Example #6
 A rocket is rising vertically at a rate of
5400 mph. An observer on the ground
is standing 20 mi from the launch point.
How fast (in radians/hr) is the angle of
elevation b/w the ground and the
observers line of sight is the rocket
increasing when the rocket is at an
elevation of 40 miles?
Example #7
 Water is being drained out of a conical
tank. Suppose the height is changing
at a rate of -.2 ft/min and the radius is
changing at a rate of -.1 ft/min. What is
the rate of change in the volume when
the radius is 1 and the height is 2?
Example #8
 A pebble is dropped into a calm pond
causing ripples in the form of circles.
The radius of the outer ripple is
increasing at a constant rate of 1 ft/sec.
When the radius is 4 feet, at what rate
is the area of the disturbed water
changing?
Example #9
 Air is being pumped into a spherical
balloon at a rate of 4.5 cubic in/min.
Find the rate of change of the radius
when the radius is 2 inches.
Example #10
 An airplane is flying on a flight path that
will take it directly over a radar station.
If the distance between the radar and
the plane is decreasing at a rate of 400
mi/hr when that distance is 10 miles,
what is the speed of the plane? The
plane is traveling at an altitude of 6
miles.
Example: From text #11
(will have variables in answer)
 Find the rate of change of the distance
between the origin and a moving point
on the graph of y=x2+1 if dx/dt = 2
cm/sec
Example #12
 Suppose a spherical balloon is inflated
at the rate of 10 cubic cm per min.
How fast is the radius of the balloon
increasing when the radius is 5 cm?
Example #13
 One end of a 13-ft ladder is on the floor
and the other end rests against a
vertical wall. If the bottom of the ladder
is drawn away from the wall at 3 ft/sec,
how fast is the top of the ladder sliding
down the wall when the bottom of the
ladder is 5 feet from the wall?
Example #14
 Water is poured into a conical paper
cup at the rate of 2/3 cubic inches per
sec. If the cup is 6 in tall and the top of
the cup has a radius of 2 in, how fast
does the water level rise when the
water is 4 in deep?
Example #15
 Pat walks at the rate of 5 ft/sec toward
a street light whose lamp is 20 ft above
the base of the light. If Pat is 6 ft tall,
determine the rate of change of the
length of Pat’s shadow at the moment
when Pat is 24 feet from the base of
the lamppost.
Example #16