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4.6 Related Rates Objective: SWBAT solve related rate problems Related Rates • When one or more values in an equation change over time, we have related rates. • We simply write equations that you might have written prior this course, then differentiate them with respect to time, t. • It is so much fun, you are going to love the process!!! That’s a Carucci guarantee. Remember that this was all with respect to x. In order to do this, we are going to have to be given information to find values for all other variables in the problem. • Related rate problems typically fall into three categories: – Pythagorean Theorem – Right-Triangle Trigonometry – Know or given formulas involving Volume, Area, or other scenarios • Answers need to be written using the correct units!!! • On a free-response question on the AP exam, units are usually worth 1 out of 9 points. This means that if you get the entire problem wrong but still include the correct units, you will at least pick up one point! Example 3: Tweety is resting in a bird house 24 feet off the ground. Using a 26 foot ladder which he leaned against the pole holding the bird house, Sylvester tries to steal the small yellow bird. Tweety’s bodyguard, Hector the dog, starts pulling the base of the ladder away from the pole at a rate of 2 ft/s. How fast is the ladder falling when it is 10 feet off the ground? We now must take the derivative with respect to time. We now have two rates: how fast x is changing and how fast y is changing. We are still missing a value for x. How can we find it? Often times in these situations you will need to substitute back into the original equation. Example 4: A police cruiser, approaching a right-angled intersection from the north, is chasing a speeding car that has turned the corner and is now moving straight east. When the cruiser is 0.6 miles north of the intersection and the car is 0.8 miles to the east, the police determine with radar that the distance between them and the car is increasing at 20 mph. If the cruiser is moving at 60 mph at the instant of measurement, what is the speed of the car? Example 5: Bugs Bunny finished his final act, and then began dancing off stage with a spotlight covering his every move. If he is moving off the stage along a straight path at a speed of 4 ft/s, and the spotlight is 20 ft away from the path, what rate is the spotlight rotating when Bugs is 15 feet from the point on the path closest to the spotlight?