Transcript Related Rates

Related Rates
Section 2.6
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Related Rates (Preliminary Notes)
If y depends on time t, then its derivative,
dy/dt, is called a time rate of change.
Usually, related rate problems will require
implicit differentiation.
Remember, dy/dt can be read as “the change
in y with respect to time”.
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Common Formulas Used in Related Rate Problems
Perimeter of rectangle:
Area of circle:
Volume of cube:
Volume of cylinder:
Volume of cone:
Volume of sphere:
Pythagorean theorem:
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Strategy For Solving Related Rate Problems
Draw a picture
Write down the given quantities and identify the
quantity to find
Relate the variables by writing an equation
Relate the rates by implicitly differentiating with
respect to time
Solve for the unknown quantity
Substitute in the known values to find the unknown
Tag on the appropriate units
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Expanding Rectangle…
Example 1: The length of a rectangle is increasing at a rate of
3ft/sec and the width is increasing at 2ft/sec. When the length is 4
ft and the width is 3ft, how fast are the perimeter, area and the
length of a diagonal increasing?
Step 1: Draw a picture
y = width
x = length
Step 2: Write down the givens and
the quantity to find
Given:
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Perimeter…
Step 2 Continued: …write down the quantity to find
a)Perimeter…
Find: The rate the perimeter is increasing when x = 4 and y = 3.
dP
?
dt x = 4
y=3
Step 3: Relate the variables
Write an equation
Step 4: Relate the rates
Implicitly differentiate with respect to time
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Perimeter…
Step 5: Solve for the unknown
Already solved in this example
Step 6: Substitute into rate equation
dP

dt
x=4
y=3
Step 7: Tag on units
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Area…
y = width
Given:
x = length
dx
 3 ft / sec
dt
dy
 2 ft / sec
dt
Step 2 Continued: …write down the quantity to find
b)Area…
Find: The rate the area is increasing when x = 4 and y = 3.
dA
?
dt x = 4
y=3
Step 3: Relate the variables
Step 4: Relate the rates
Step 5: Solve for the unknown
Already solved in this example
Step 6: Substitute into rate equation
dA

dt x = 4
y=3
Step 7: Tag on units
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Diagonal Length…
D
y
Step 4: Relate the rates
x
Step 2 (cont) Find:
The rate the diagonal is
increasing when x = 4 and y = 3.
dD
?
dt
Step 5: Solve for the unknown
dD

dt
x=4
y=3
Step 3: Relate the variables
Step 6: Substitute into rate equation
dD

dt x = 4
y=3
D=
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Increasing the Area of a Circle
Example 2 The radius of a circle is increasing at the rate of 5 inches
per minute. At what rate is the area increasing when the radius is 10
inches?
1) Given:
2) Find:
3) Relate the variables (formula):
4) Relate the rates (implicit differentiation):
5) Substitute into rate equation:
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Blowing Up a Balloon
Example 3: Air is being pumped into a spherical balloon at a
rate of 4.5 cubic feet per minute. Find the rate of change of the
radius when the radius is 2 feet.
1) Given:
4) Relate the rates (implicit differentiation)
R
5) Solve for the unknown
2) Find:
dR
?
dt
R= 2
3) Relate the variables (formula)
6) Substitute into rate equation:
dR

dt
R= 2
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The Sliding Ladder…part a)
Example 4: A 5 meter-long ladder is leaning against the side of a house. The
foot of the ladder is pulled away from the house at a rate of 0.4 m/s.
a) Determine how fast the top of the ladder is descending when the foot of the
ladder is 3 meters from the house.
4) Relate the rates (implicit differentiation)
1) Given:
5
y
x
dx

dt
2) Find:
dy
?
dt x = 3
3) Relate the variables (formula)
5) Solve for the unknown
6) Substitute into rate equation:
dy

dt
x=3
y=
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The Sliding Ladder…part b)
Example 4: b) Determine the rate at which the angle between the ladder and the
wall is changing when the base of the ladder is 3 m from the house.
1) Given:

5
y
dx

dt
4) Relate the rates (implicit differentiation)
x
2) Find:
5) Solve for the unknown
d
?
dt x = 3
3) Relate the variables (formula)
6) Substitute into rate equation:
d

dt
x=3
y =4
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Volume of a Conical Sand Pile
Example 5: Sand is falling off a conveyor onto a conical pile at the rate of
15 cubic feet per minute. The diameter of the base of the cone is
approximately twice the altitude. At what rate is the height of the pile
changing when it is 10 feet high?
1) Given:
h
2r
2) Find:
dh
?
dt h = 10
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Homework
Section 2.6 page 149 #3, 5, 15, 19, 20, 23, and 27
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