Related Rates

1. A pebble is dropped into a calm pond causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate of 1 foot per second. When the radius is 4 feet, at what rate is the total area of the disturbed water changing? ANS: 8 π ft2/sec.
2. A ladder is 25 ft long and leaning against a vertical wall. The bottom of the ladder is pulled horizontally away from the wall at 3 ft/s. How fast the top of the ladder sliding down the wall when the bottom is 15 ft from the wall? ANS: –2.25 ft/s.
3. Two cars, one going due east at the rate of 90 km/h and the other going due south at the rate of 60 km/h, are travelling toward the intersection of the two roads. At what rate are the cars approaching each other at the instant when the first car is 0.2 km and the second car is 0.15 km from the intersection? ANS: –108 km/h.
Here is an applet describing this situation: http://www.usna.edu/MathDept/cdp/relatedrates/rates.html
4. Suppose in a certain market that x thousands of crates of oranges are supplied daily when p dollars is the price per crate, and the supply equation is
px – 20p – 3x + 105 = 0
If the daily supply is decreasing at the rate of 250 crates per day, at what rate is the price changing when the daily supply is 5000 crates? ANS: –$0.05 per crate.
5. An airplane is flying west at 500 ft/s at an altitude of 4000 ft and a searchlight on the ground lies directly under the path of the plane. If the light is kept on the plane, how fast is the searchlight revolving when the horizontal distance of the plane from the searchlight is 2000 ft due east? ANS: 0.1 degree per second.
Here is an applet describing an analogous situation: http://math.dartmouth.edu/~klbooksite/2.17/217.html
6. At a sand and gravel pit, sand is falling off a coveyor and onto a conical pile at the rate of 10 cubic feet per minute. The diameter of the base of the cone is approximately three times the altitude. At what rate is the height of the pile changing when the pile is 15 feet high? ANS: 8/(405π) ft/min.
Steps:
  1. Read problem carefully
  2. Draw diagram showing relationship of variables
  3. Define variables (usually each variable will be dependent on time, t)
  4. Write down known numerical derivatives
  5. Identify the unknown (i.e., the quantity for which you need to solve)
  6. Write down an equation that relates the variables listed in #3
  7. Implicitly differentiate the equation with respect to t
  8. Substitute known quantities into the equation
  9. Solve and write answer in a complete sentence

Note: Sometimes, in step #8, when you substitute the known quantities into the equation, you may have a remaining variable with no value. Usually, you will need to return to step #6 to calculate a value for the remaining variable.


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