Chapter 4.6
Related Rates
1
ex: A hot­air balloon rising straight up from a level field is tracked by a range finder 500 ft. from the lift­off point. At the moment the range finder's elevation angle is , the angle is increasing at the rate of 0.14 radians per min. How fast is the balloon rising at that moment?
π
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Steps:
1. Identify rate of change of what you know and want.
2. Picture! Don't freeze it!
3. Write eq'n with variables you want and know. The eq'ns will depend on what you're given in each problem.
4. Differentiate implicitly with respect to time.
5. Plug in values (always do after differentiating).
6. Interpret sol'n.
2
ex: A police cruiser, approaching a right­angled intersection from the north, is chasing a speeding car that has turned a corner and is now moving straight east. When the cruiser is 0.6 mi. north of the intersection and the car is 0.8 mi 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 the measurement, what is the speed of the car?
3
ex: Air is being pumped into a spherical balloon so that 3
its volume increases at a rate of 100 cm
/sec. How fast is the radius of the balloon increasing when the diameter is 50 cm?
4
ex: A 10 ft. long ladder rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft/sec, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft. from the wall?
5
ex: Car A is traveling west at 50 mph and car B is traveling north at 60 mph. Both are headed for a 4­way stop intersection of the two roads. At what rate are the cars approaching each other when car A is 0.3 miles and car B is 0.4 miles from the intersection?
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ASSIGNMENT
p. 251 #8, 9, 11a, 19a
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ASSIGNMENT­ Stewart Ch. 3.9 #11, 31
11. Two cars start moving from the same point. One travels south at 60mph and the other travels west at 25mph. At what rate is the distance between the cars increasing two hours later?
31. A ladder 10 ft. long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a speed of 2 ft./sec, how fast is the angle between the top of the ladder and the wall changing when the angle
π
is rad?
4
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