1. Bell Ringer.
2. Related rates problems involve two rates which are related to each other.
How do you find instantaneous rate?
Answer: By taking the derivative with respect to time.
3. Example:
An off-shore tanker begins leaking oil so that the radius of the circle is changing at the rate of 2 miles/hour. How fast is the area of the oil spill changing when the radius is 6miles?
4. Strategy for solving related rates problems
(1) Draw and label a diagram.
(2) Write down what you wish to solve for (express it in terms of a variable). e.g., dh/dt
(3) Identify other variables in the problem.
(4) Write equations that relate the variables.
(5) Use substitution to obtain one equation involving the known quantities.
(6) Take the derivative of each side with respect to time.
(7) Solve for the desired rate.
(8) Make numerical substitutions.
5. Example: Ladder Problem
Draw this on the blackboard and show that the rates are not the same.
A 26-foot ladder is leaning against a wall. If it is being pulled away from the wall at a rate of 4 ft/sec, How fast is the top of the ladder sliding down the wall when the foot of the ladder is 10 feet from the wall?
Answer: -5/3 ft/sec
Example: Kite Problem
A kite is at an altitude of 75 feet and moving horizontally at 12 ft/sec. How fast is the string being paid out when the kite is 100 feet away (measured along the string)?
Answer is 7.94 ft/sec