6.5: Related Rates

Objective:To use implicit differentiation to relate the rates in which 2 things are changing, both with respect to time.

Air is pumped into a spherical balloon so that its volume increases at a rate of 100cm3/s. How fast is the radius increasing when the diameter is 50cm?

Related Rates Guidelines1. Make a sketch. Label all sides in terms of variables, even if you are given the

actual values of the sides.2. Make a list of variables. Separate them into variables that are constant (never

change) and variables that are changing (variables that are a given value only at a certain point in time). Rates (recognized by “increasing”, “decreasing” , etc.) are derivatives with respect to time. They can go into either category. Be aware if the rate is positive or negative (increasing vs. decreasing over time).

3. Find an equation which ties your variables together. If it is an area problem, you need an area equation. If it is a right triangle, the Pythagorean formula may work, etc.

4. Plug in any variable that is constant. NEVER plug in a variable that is changing.

5. Use implicit differentiation to differentiate you equation with respect to time, t.

6. Plug in all variables and known rates. Solve for the unknown rate. 7. Label your answers in terms of the correct units ( very important), and be

sure you answered the question asked.

Things to remember……

Words such as “rate” and “speed” are code words for derivative, where the underlying variable is time, t.

Be aware of positive vs. negative rates of change. Is “something” increasing or decreasing over time?

If “something” is constant or does not change over time, the rate of change, or derivative, is 0.

Ex. 1: Air is pumped into a spherical balloon so that its volume increases at a rate of 100cm3/s. How fast is the radius increasing when the diameter is 50cm?

Ex. 2: A 10 ft ladder is resting against a vertical wall when suddenly it starts to slide, with the bottom moving away from the wall at a rate of 1 ft/s. How fast is the top sliding down when the base is 6 feet from the wall?

Ex. 3 An inverted circular cone that has a radius of 2m and a height of 4 m is being filled with Slime at a rate of 2 m3/min. How fast is the height of the Slime increasing when it is 3m deep?

Ex. 4: A trough of water is 8 meters long and its ends are in the shape of isosceles triangles whose width is 5m and height is 2 m. Water is being pumped in at a constant rate of 6 m3/sec. At what rate is the height of the water changing when the water has a height of 1.2 m?

Ex. 5: You’re meeting your friend for breakfast at Percy’s in Kingston. If you’re driving south at 60mph and your friend is driving east at 50mph, how fast will you be approaching them when you’re 0.4 miles away and they’re 0.3 miles away?