RELATED RATES

The Hoover Dam

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