Section 2-6 Related Rates
Definition:
When two or more related variables
are changing with respect to time
they are called related rates
Derivative = rate of change
• When we say “rate of change” without
specifying what it's with respect to, we
mean “rate of change with respect to
time”.
– example: velocity- change in distance with respect to time
• If the variable is different from time use
implicit differentiation
• Unless otherwise specified assume rate of
change means with respect to time
1) Differentiate each with respect to t.
A  r
2
4 3
V  r
3
c  a b
2
2
y
tan  
50
2
2) Suppose x and y are both differentiable
functions with respect to time, t, and are related
by the equation y = x3 + 5 find dy/dt when x =
3, given that dx/dt = 2 when x = 3.
~ Differentiate both sides of the equation with respect
to t using the chain rule.
3) Suppose x and y are both differentiable
functions with respect to time, t, and are
related by the equation x2y = 2. Find dy/dt
when x = 3, given that dx/dt = 2 when x =
3.
Helpful suggestions for solving Related Rates
• Identify all known and unknown quantities. Make
a sketch and label it.
• Find an equation that describes the relationship
between the variables. Common equations
include the Pythagorean Theorem, Area of circle,
Volumes of known geometric shape, SOH-CAHTOA, similar triangles, etc.
• Differentiate both sides of the equation with
respect to time.
• Substitute all known quantities and rates into the
equation and solve.
4) A 13 – foot ladder is leaning against the
wall of a house. The base of the ladder
slides away from the wall at a rate of 0.75
feet per second. How fast is the top of the
ladder moving down the wall when the base
is 12 feet from the wall?
5) Oil spills into a lake in a circular pattern. If
the radius of the circle increases at a
constant rate of 3 feet per second, how fast
is the area of the spill increasing at the end
of 30 minutes.
6) A baseball diamond is a square with side
90 feet. A batter hits the ball and runs
toward first base with a speed of 24 feet
per second. At what rate is his distance
from second base decreasing when he is
halfway to first base?
7) A weather balloon is released 50 feet
from an observer. It rises at a rate of 8 feet
per second. How fast is the angle of
elevation changing when the balloon is 50
feet high?
Ɵ
8) A man 6 feet tall is walking toward a lamppost
20 feet high at a rate of 5 feet per second. The
light at the top of the lamppost (20 feet above the
ground) is casting a shadow of the man. At what
rate is the tip of his shadow moving and at what
rate is the length of his shadow changing when
he is 10 feet from the base of the lamppost?
9) Two cars start moving from the same
point at noon. One travels south at 60 mi/h
and the other travels west at 25 mi/h. At
what rate is the distance between the cars
increasing at 2:00 pm?
10) Water is being poured into a conical
reservoir at the rate of pi cubic feet per
second. The reservoir has a radius of 6
feet across the top and a height of 12
feet. At what rate is the depth of the water
increasing when the depth is 6 feet?
11) Air is being pumped into a spherical ball
at a rate of 5 cubic centimeters per minute.
Determine the rate at which the radius of
the ball is increasing when the diameter of
the ball is 20 cm.
12) 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?
15 ft
13) A family and a guy are 50 feet apart. The
family starts walking north at a rate so that the
angle between them is changing at a constant
rate of 0.01 radians per minute. At what rate is
the distance between them changing when   0.5
radians?

14) Two people on bikes are separated by 350
meters. Person A starts riding north at a rate of
5m/sec and 7 minutes later Person B starts riding
south at 3m/sec. At what rate is the distance
separating the two people changing 25 minutes
after person A starts riding?
A
B
Homework
Page 154
Problems # 1, 3,11,13,17, 20, 22, 25, 29,
31 ,32, 33, 43, and 44