Section 3.9: Related Rates

**Guidelines for Solving Related-Rate Problems:**

- If possible, draw a picture.
- Identify all given quantities and all quantities to be determined.
- Write an equation involving the variables whose rates of change either are given or are to be determined.
- Implicitly
differentiate both sides of this equation with respect to
*t*. - Substitute all known values, including know rates of change, into the resulting equation and then solve for the required rate of change.

**Example:** A nugget is dropped into a calm pond,
causing ripples in the form of concentric circles. The radius *r* of the outer ripple is increasing
at a constant rate of 2 ft/sec.
When the radius is 3 feet, at what rate is the total area *A* of the outer ripple changing?

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**Example:** A nugget is flying on a flight path 3
miles above the ocean that will take it directly over an island. If the distance between the nugget and
the island is decreasing at a rate of 200 miles per hour when the distance between
them is 5 miles, what is the speed of the nugget?

**Example:** A nugget is rising at a rate of 15 feet
per second when the nugget is 50 feet off the ground. A photographer is standing on the ground 100 feet from the
take-off site. If the photographer
keeps the nugget in his/her sight, what is the change in the photographerâ€™s
angle of elevation when the nugget is 50 feet off the ground?