Calculus 2'7 Related Rates

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Calculus2.7 Related Rates
Mrs. Kessler
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In algebra we study relationships among variables

• The volume of a sphere is related to its radius.
  
• The sides of a right triangle are related by
  Pythagorean Theorem.
• The angles in a right triangle are related to the
  sides.

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In calculus we study relationships between the
rates of change of variables.
For example, how is the rate of change of the
radius of a sphere related to the rate of change
of the volume of that sphere?
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Solving Related Rates Equations

• Read the problem at least three times and draw a
  picture
• Identify all the given quantities and the
  quantities to be found (these are usually rates.)
• Draw a sketch and label, using unknowns when
  necessary.
• Write an equation (formula) that relates the
  variables.
• Assume all variables are functions of time and
  differentiate wrt time using the chain rule. The
  result is called the related rates equation.
• Substitute the known values into the related
  rates equation and solve for the unknown rate.

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Finding derivative with respect to t
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If the radius changes 0.1cm/sec (a very small
amount) how fast does the volume change?
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A hot-air balloon rising straight up from a level
field is tracked by a range finder 500 ft from
the liftoff point. The angle of elevation is
increasing at the rate of 0.14 rad/min. How fast
is the balloon rising when the angle of elevation
is is ?/4?
Given
Find
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Ex 2 contd
A hot-air balloon rising straight up from a level
field is tracked by a range finder 500 ft from
the liftoff point. At the moment the range
finders elevation angle is ?/4, the angle is
increasing at the rate of 0.14 rad/min. How fast
is the balloon rising at that moment?
What equation has all the knowns and unknowns in
it?
Now take the derivative, implicitly.
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Figure 2.44 Figure for Example 4.
Ex 3
A police cruiser, approaching a right angled
intersection from the north is chasing a speeding
car that has turned the corner and is now moving
straight east. The cruiser is moving at 60 mph
and the police determine with radar that the
distance between them is increasing at 20 mph.
When the cruiser is .6 mi. north of the
intersection and the car is .8 mi to the east,
what is the speed of the car?
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Figure 2.44 Figure for Example 4.
Ex 3 contd
A police cruiser, approaching a right angled
intersection from the north is chasing a speeding
car that has turned the corner and is now moving
straight east. The cruiser is moving at 60 mph
and the police determine with radar that the
distance between them is increasing at 20 mph.
When the cruiser is .6 mi. north of the
intersection and the car is .8 mi to the east,
what is the speed of the car?
Given
Find
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Ex 3 contd
What equation has all the knowns and unknowns in
it?
Now take the derivative, implicitly
Given
Now sub in the given information
, s 1
What is s?
Use Pythagorean Thm.
solve
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Ex 4
Water is draining from a cylindrical tank at 3
cubic feet /second. How fast is the surface
dropping in inches per second? The radius is 15
in.
What equation has all the knowns and unknowns in
it?
(We need a formula to relate V and h.
(Since the radius is not changing, we treat it
like a constant.
(r is a constant.)
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Ex 4
Water is draining from a cylindrical tank at 3
cubic feet/ second. How fast is the surface
dropping when r 15 in
NOTE, the problem must be done in same units of
measure. Since r is in inches and we have already
used that numberin the problem we have to change
the cu.ft. to cu. inches.
How many cubic inches in a cubic foot.?
1728
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Truck A travels east at 40 mi/hr. Truck B travels
north at 30 mi/hr.
Truck Problem
Ex 6
How fast is the distance between the trucks
changing 6 minutes later?
What equation has all the knowns and unknowns in
it?
B
z
Now take the derivative, implicitly
y
A
x
Now sub in the given information
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Ex 6
We must find x, y, and z at the specific time
Note 6 min 1/10 hr.
B
Eastbound
Northbound
A
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Truck Problem
Truck A travels east at 40 mi/hr. Truck B travels
north at 30 mi/hr.
Ex 6
How fast is the distance between the trucks
changing 6 minutes later?
B
A
Will it always be that way? Try different speeds
and times and see what happens.
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Ex 5
Water runs into a conical tank at the rate of 9
ft3/min. The tank stands point down and has a
height of 10 ft and a base of radius 5 ft. How
fast is the water level rising when the water is
6 ft. deep?
Given
Find
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Ex. 5 Water runs into a conical tank at the
rate of 9 ft3/min. The tank stands point down and
has a height of 10 ft and a base of radius 5 ft.
How fast is the water level rising when the water
is 6 ft. deep?
What equation has all the knowns and unknowns in
it?
If we differentiate implicitly, we will have
three rates, dV/dt, dh/dt, and dr/dt .
r
We cannot determine dr/dt. So we have to
eliminate r in the equation. Similar triangles
again.
h
Sub in for r in the volume formula
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Ex 5
Water runs into a conical tank at the rate of 9
ft3/min. The tank stands point down and has a
height of 10 ft and a base of radius 5 ft. How
fast is the water level rising when the water is
6 ft. deep?
Given
Solve for dh/dt
Find
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Ex. 6 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.
y distance from the top of the ladder to the
ground x distance from the bottom of the
ladder to the wall 
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Ex. 6 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.
y distance from the top of the ladder to the
ground x distance from the bottom of the
ladder to the wall 
Set up a related equation.
8
y
What is y when x 4?
x
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Ex. 6 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.
What is y when x 4?
8
y
x
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Ex. 7 A fish is reeled in at a rate of 1 foot
per second from a point 10 feet above the water. 
At what rate is the angle between the line and
the water changing when there is a total of 25
feet of line out?
x the amount of line out z the horizontal
distance from the fish to the line ? is the angle
between the fish and the water.
What equation related the variables?
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A fish is reeled in at a rate of 1 foot per
second from a point 10 feet above the water.  At
what rate is the angle between the line and the
water changing when there is a total of 25 feet
of line out?
Take the derivative implicitly.
What is cos ??
First find z. Use Pythag.
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A fish is reeled in at a rate of 1 foot per
second from a point 10 feet above the water.  At
what rate is the angle between the line and the
water changing when there is a total of 25 feet
of line out?
First find z. Use Pythag.
What is cos ??
Approximately 1º/sec.
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HW1.

• The radius of a circle is increasing at the rate
  of 2 cm/min. Find the rate of change of the area
  when r 6 and also when r 24

b. 96?
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• 2. A spherical balloon is inflated with gas at
  the rate of 500cc per minute. How fast is the
  radius of the balloon increasing when the radius
  is 30 cm.? 60cm.?

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HW 3

• Air traffic control is flying a plane at an
  altitude of 6 miles and passes directly over a
  radar antenna. When the plane is 10 miles away
  (S 10), the radar detects that the distance s
  is changing at a rate of 240 mph. What is the
  speed of the plane?

6
10
s
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HW 4

• Water is falling on a surface wetting a circular
  area that is expanding at a rate of 3 mm2/sec.
  How fast is the radius of the wetted arwa
  expanding when the radius is12 mm.

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HW 5
A man 6 ft. tall walks at a rate of 2 ft./sec
away from a lamp post that is 13 ft. high. At
what rate is the length of his shadow changing
when he is 25 ft. away from the lamp post.

• This one is difficult

We want
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Know
x
y
x y
What is the relationship between x and y?
Use similar triangles.
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HW 5
A man 6 ft. tall walks at a rate of 2 ft./sec
away from a lamp post that is 13 ft. high. At
what rate is the length of his shadow changing
when he is 25 ft. away from the lamp post.
The 25 did not matter.
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6
x
y
x y