Section 2.6
RELATED RATES
WHAT EXACTLY IS A “RELATED RATE”?

a “related rate” problem occurs when two or
more variables are changing with respect to
TIME in a given situation (means that t is the
“chief” variable and you therefore differentiate
with respect to t)
PROCEDURE IN SOLVING THESE TYPES OF
PROBLEMS…

pg. 150, gray box entitled “Guidelines For
Solving Related-Rate Problems” (copy into your
notebook)
TYPE #1: EXPANDING/SHRINKING OBJECT

A pebble is dropped into a calm pond, causing
ripples in the form of concentric circles. The
radius r of the outer ripple is increasing at a
constant rate of 2 ft/sec. When the radius is 5
feet, at what rate is the total area A of the
disturbed water changing?
ANOTHER EXAMPLE:

Air is being left out of a spherical balloon at a
rate of 3.5 cm3/min. Determine the rate at
which the radius of the balloon is decreasing
when the diameter of the balloon is 20 cm.
AND ONE MORE…

Oil spilled from a ruptured tanker spreads out
in a circle whose area increases at a constant
rate of 6 mi2/hr. How fast is the radius of the
spill increasing when the area is 9 square
miles?
TYPE #2: THE “SLIPPING LADDER”

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.
ANOTHER EXAMPLE…

A 25-foot ladder is leaning against a vertical
wall. The floor is slightly slippery and the foot
of the ladder slips away from the wall at the
rate of 0.2 inches per second. How fast is the
top of the ladder sliding down the wall when
the top is 20 feet above the floor?
AND ONE MORE…

A 15 foot ladder is resting against the wall. The
bottom is initially 10 feet away from the wall
and is being pushed towards the wall at a rate
of .25 ft/sec. How fast is the top of the ladder
moving up the wall 12 seconds after we start
pushing?
TYPE #3: THE FILLING (OR UNLOADING)
CONE/TROUGH

A conical water tank with vertex down has a
radius of 10 feet at the top and is 24 feet high.
If water flows out of the tank at a rate of 20
cubic feet per minute, how fast is the depth of
the water decreasing when the water is 16 feet
deep?
ANOTHER ONE…

Water is flowing into an inverted right circular
cone at a rate of 4 cubic inches per minute.
The cone is 16 inches tall and its base has a
radius of 4 inches. At the moment the water
has a depth of 5 inches, how fast is the radius
at the surface of the water increasing?
AND ONE MORE…
A tank of water in the shape of a cone is leaking
water at a constant rate of 2 cubic feet per
hour. The base radius of the tank is 5 ft and the
height of the tank is 14 ft.
 (a) At what rate is the depth of the water in the
tank changing when the depth of the water is 6 ft?


(b) At what rate is the radius of the top of the water
in the tank changing when the depth of the water
is 6 ft?
TYPE #4: THE “PUSHING” PROBLEM

A person is pushing a box up a 30 foot long ramp
at the rate of 4.5 feet per second. The ramp rises
to a height of 7 feet. How fast is the box rising?

A package, while being pushed along a conveyor
belt “ramp”, is being risen at a rate of 3.5 feet per
second. If the conveyor belt is 40 feet long and
rises to a height of 5 feet, how fast is the package
being pushed horizontally along the belt?