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?
Example of a Related Rate:
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 1: Read the problem carefully.
Step 4: Write an equation relating the quantities.Step 5: Implicitly differentiate both sides of the equation with respect to time, t.
Step 7: Solve for the unknown.Step 8: Check your answers to
see that they are reasonable.
Step 6: Substitute values into the derived equation.
CAUTION: Be sure to include units of measurement in your answer.
CAUTION: Be sure the units of measurement match throughout the problem.
The table below lists examples of mathematical models involving rates of change. Let’s 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.
C = 2 rA = r2
V = 4/3r3
SA = 4 r2
a2 + b2 = c2
a
b
cr
r
h
V = r2h
r
h
V = 1/3 r2h A = 1/2 bh
h
b
30
60
x
x/2
x/2√3
r
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?
Let’s try:
What are we
trying to find?
What variable can we assign
this unknown?
dAdt
=?
What formula
can I use?
A = r2
How can I get dA/dt
out of that
formula?
dA/dt = 2r dr/dt
Substitute in what
you know!
dA/dt = 2(30 m)(1 m/s)dA/dt = 60 m2/s
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
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.
r
What are we trying to find?
What variable
can we use to define
the unknown?
dddt
=?
What formula can we use?
SA = 4r2
How can we get dd/dt out of this formula?
We have to rewrite this
formula so that it has a
diameter instead of a
radius…
SA = 4(1/2d)2
Can you finish from
here?
Let’s 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
Let’s 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
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
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