Related Rates

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Related Rates
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The Hoover Dam
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Example of a Related Rate
Oil spills from a ruptured tanker and spreads in
a circular pattern. If the radius of the oil
spill increases at a constant rate of 1m/s, how
fast is the area of the spill increasing when the
radius in 30m?
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Step 1 Read the problem carefully.
Step 2 Draw a picture to model the situation.
Step 3 Identify variables of the known and the
unknown. Some variables may be rates.
Step 4 Write an equation relating the
quantities.
Step 5 Implicitly differentiate both sides of
the equation with respect to time, t.
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Step 6 Substitute values into the derived
equation.
Step 7 Solve for the unknown.
Step 8 Check your answers to see that they are
reasonable.
CAUTION Be sure the units of measurement match
throughout the problem.
CAUTION Be sure to include units of measurement
in your answer.
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The table below lists examples of mathematical
models involving rates of change. Lets
translate them into variable expressions
Verbal Statement Mathematical Model
Water is being pumped into a swimming pool at a rate of 10 cubic meters per hour.
The velocity of a car is 50 miles per hour
The length of a rectangle is decreasing at a rate of 2 cm/sec.
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Geometry Formula Review
h
r
r
r
V ?r2h
C 2 ?r A ?r2
V 4/3?r3 SA 4 ?r2
a2 b2 c2
30?
x
x/2?v3
h
h
60?
r
b
x/2
A 1/2 bh
V 1/3? r2h
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Lets try
Oil spills from a ruptured tanker and spreads in
a circular pattern. If the radius of the oil
spill increases at a constant rate of 1m/s, how
fast is the area of the spill increasing when the
radius is 30m?
What formula can I use?
Substitute in what you know!
What are we trying to find?
How can I get dA/dt out of that formula?
What variable can we assign this unknown?
dA/dt 2?r dr/dt
dA dt
?
dA/dt 2?(30 m)(1 m/s)
A ?r2
dA/dt 60? m2/s
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Your turn
A child throws a stone into a still pond causing
a circular ripple to spread. If the radius
increases at a constant rate of 1/2m/s, how fast
is the area of the ripple increasing when the
radius of the ripple is 20 m?
Answer 20? m2/s or 62.8 m2/s
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The process might get more involved.
If a snowball (perfect sphere) melts so that its
surface area decreases at a rate of 1 cm2/min,
find the rate at which the diameter decreases
when the diameter is 10 cm.
We have to rewrite this formula so that it has a
diameter instead of a radius
What variable can we use to define the unknown?
What formula can we use?
Can you finish from here?
How can we get dd/dt out of this formula?
What are we trying to find?
r
dd dt
?
SA 4?r2
SA 4?(1/2d)2
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Lets try more
Two cars start moving from the same point. 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 2 hours later?
Answer 65 mi/h
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Lets try more
A ladder 10 ft. long rests against a vertical
wall. If the bottom of the ladder slides away
from the wall at a rate of 1 ft/s, how fast is
the top of the ladder sliding down the wall when
the bottom of the ladder is 6 ft. from the wall?
Answer -3/4 ft/s
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A trough is 10 ft long and its ends are in the
shape of isosceles triangles that are 3 ft across
at the top and have a height of 1 feet. If the
trough is filled with water at a rate of 12 feet
cubed per minute, how fast is the water level
rising when the water is half a foot deep?
Answer 4/5 ft/min