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Calculus 2.6 Guided Notes – Related Rates
2.6 Related Rates
Objectives
 Find a related rate.
 Use related rates to solve real-life problems.
Finding Related Rates
You have seen how the Chain Rule can be used to find dy/dx implicitly.
Another important use of the Chain Rule is to find the rates of change of two or more related variables
that are changing with respect to time.
For example, when water is drained out of a conical tank (see Figure 2.33), the volume V, the radius r,
and the height h of the water level are all functions of time t.
Knowing that these variables are related by the equation
you can differentiate implicitly with respect to t to obtain the related-rate equation
1) Find dV/dt
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Calculus 2.6 Guided Notes – Related Rates
EXAMPLE PROBLEM
Suppose x and y are both differentiable functions of t and are related by the equation y = x2 + 3.
Find dy/dt when x = 1, given that dx/dt = 2 when x = 1.
Problem Solving with Related Rates
A pebble is dropped in a calm pond, causing ripples in the from of concentric circles. The radius r of the
outer ripple is increasing at a rate of 1 foot per second. When the radius is 4 feet, at what rate is the
total area A of the disturbed water changing?
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Calculus 2.6 Guided Notes – Related Rates
The table below lists examples of mathematical models involving rates of change.
Practice Problems
Oil spilling from a ruptured tanker spreads in a circle on the surface of the ocean. The area of the spill
increases at a rate of 9π m²/min. How fast is the radius of the spill increasing when the radius is 10 m?
A spherical balloon is inflated so that its radius (r) increases at a rate of 2r cm/sec. How fast is the
volume of the balloon increasing when the radius is 4 cm?
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Calculus 2.6 Guided Notes – Related Rates
A 6 ft tall person is walking away from a 20 ft tall lamppost at a rate of 5 ft/sec. Assume the scenario can
be modeled with right triangles. At what rate is the length of the person's shadow changing when the
person is 16 ft from the lamppost?
1. Suppose the radius of a spherical balloon is shrinking at ½ in/min. How fast is the volume
decreasing when the radius is 4 inches?
2. A person is pushing a box up a ramp at 3 ft/sec. If the ramp is 20 ft. long and 5 ft. high, how fast
is the box rising up the ramp?
3. Car A is traveling west at 50 mi/hr. and car B is traveling north at 60 mi/hr. Both are headed for
the intersection of the two roads. At what rate are the cars approaching each other when car A
is 0.3 mi. and car B is 0.4 mi. from the intersection?
Name:_________________________________
Calculus 2.6 Guided Notes – Related Rates
Name:_________________________________
Calculus 2.6 Guided Notes – Related Rates