Calculus Section 3.10 Related Rates

An 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.

Ex. 1) Assume x and y are both differentiable functions of t and are related by the equation . a) Find when given that . Steps: 1. Differentiate all terms with respect to t. 2. Substitute the given values. 3. Isolate the unknown.

b) Find when given that .

Assignment: Assume that x and y are both differentiable functions of t and find the indicated values. 1. find when given 2. find when given 3. find when given 4. find when given 5. find when given Calculus Section 3.10 Related Rates Day 2 Ex. 1) Air is being pumped into a spherical balloon so that its volume increases at a rate of 100. How fast is the radius of the balloon increasing when the diameter is 50 cm? What are you looking for: What do you know: The rate of increase of the radius: The rate of increase of volume: when

The formula for the volume of sphere relates volume to radius.

Ex. 2) A ladder 10 feet long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 feet from the wall? Given find when . What is the relationship between x and y? y

Assignment: Worksheet Calculus Section 3.10 Related Rates Day 3 Ex. 1) Oil spilled from a ruptured tank spreads in a circle whose area increases at a constant rate of . How fast is the radius of the spill increasing when the radius is 2 miles?

Ex. 2) A water spills spreads in a circular pattern. If the radius of the spill increases at a rate of 2 in/s, how fast is the circumference of the circle increasing?

Assignment: Quiz Review