Quote of the Day:
"The calculus is the greatest aid we have to the
application of physical truth in the broadest sense
of the word." -- W. F. Osgood
Objectives:
The student will solve related rates problems.
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
6. Assignment:
Read p. 217 - 220
p. 221 (1a, 7, 10, 13, 15, 17)
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