MAT 1234
Calculus I
Section 2.8
Related Rates
http://myhome.spu.edu/lauw
Next..
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WebAssign 2.8. Due Next Monday (Difficulty
level *****) to give you more time. Please do
not wait until Monday afternoon.
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(WebAssign 2.9 is also due Monday.)
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Be sure to do it ASAP. Tutors are available
today and tomorrow after class.
Write down your solutions carefully!!!
One of these type of questions will be on the
second exam.
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Room Change Next Thursday
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Switch to 245
Preview
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Define Related Rates
How to solve word problems involving
Related Rates
Related Rates
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 
 
d 2
dx d 5
4 dy
x  2x ;
y  5y
dt
dt dt
dt
Example 1
GO NUTS!
Example 1
Example 1
A rock is thrown into a still pond and
causes a circular ripple. If the radius of the
ripple is increasing at 3 feet per second,
how fast is the area changing when the
radius is 5 feet?
Step 1 Draw a diagram
A rock is thrown into a still pond and
causes a circular ripple. If the radius of the
ripple is increasing at 3 feet per second,
how fast is the area changing when the
radius is 5 feet?
Step 2: Define the variables
A rock is thrown into a still pond and
causes a circular ripple. If the radius of the
ripple is increasing at 3 feet per second,
how fast is the area changing when the
radius is 5 feet?
Step 3: Write down all the information
in terms of the variables defined
A rock is thrown into a still pond and
causes a circular ripple. If the radius of the
ripple is increasing at 3 feet per second,
how fast is the area changing when the
radius is 5 feet?
Step 4: Set up a relation
between the variables
Step 5: Use differentiation to
find the related rate
ft 2 / s
Expectations
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In the quizzes and exams, you are
expected to include these 5 steps in your
solutions.
Example 2
A 26-foot ladder is placed against a wall. If the top of
the ladder is sliding down the wall at 2 feet per second,
at what rate is the bottom of the ladder moving away
from the wall when the bottom of the ladder is 10 feet
away from the wall?
Example 2
A 26-foot ladder is placed against a wall. If the top of
the ladder is sliding down the wall at 2 feet per second,
at what rate is the bottom of the ladder moving away
from the wall when the bottom of the ladder is 10 feet
away from the wall?
Everyone, try step 1 and 2!
Step 1 Draw a diagram
A 26-foot ladder is placed against a wall. If the top of
the ladder is sliding down the wall at 2 feet per second,
at what rate is the bottom of the ladder moving away
from the wall when the bottom of the ladder is 10 feet
away from the wall?
Step 2: Define the variables
A 26-foot ladder is placed against a wall. If the top of
the ladder is sliding down the wall at 2 feet per second,
at what rate is the bottom of the ladder moving away
from the wall when the bottom of the ladder is 10 feet
away from the wall?
Remark: Do not define more variables
than necessary.
Step 3: Write down all the information
in terms of the variables defined
A 26-foot ladder is placed against a wall. If the top of
the ladder is sliding down the wall at 2 feet per second,
at what rate is the bottom of the ladder moving away
from the wall when the bottom of the ladder is 10 feet
away from the wall?
Step 4: Set up a relation
between the variables
Step 5: Use differentiation to
find the related rate
ft / s
x  10, y  24
Review: Similar Triangles
Two triangles are similar if and only if
one of the following 2 conditions are
satisfied
1. Their corresponding angles are
the same.
2. The ratio of their corresponding
sides are the same.
Review: Similar Triangles
In particular:
If the corresponding angles are the
same, then the ratio of their
corresponding sides are the same.
Please wait…
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We are going to walk through some of
the main key points in your classwork.
Please do not start your classwork now,
not even drawing the diagrams.
Example 3 (Classwork)
A street light is mounted at the top of a 12ft-tall pole. A 6-ft-tall man walks away from
the pole with a speed of 4ft/s along a
straight path. How fast is the tip of his
shadow moving when he is 35 ft from the
pole?
Example 3
4 ft/s
Wall
12 ft
Man
6 ft
35 feet
???? ft/s
Example 3
Remark: Do not define more variables
than necessary.
For example, it is not necessary to
define a variable for the length of the
shadow.
12
6
x
z
Example 3
Remark: Do not define more variables
than necessary.
For example, it is not necessary to
define a variable for the length of the
shadow.
12
6
z-x
x
z
Example 3
dx
 4ft/s
dt
dz
?
dt x 35
12
6
z-x
x
z
Hint
Use similar triangles to
find a relation between
x and z.
12
Solve z in terms of x.
6
z-x
x
z
The Answers
It turns out that in this
problem, the answer is
independent of the fact
dz
 ???(constant) that x=35. This means
dt
that the tip of the
dz
shadow is moving at a
 ??? ft/s
dt x 35
constant rate.