1. Related Rates

2. I. Procedure A.) State what is given and what is to be found! Draw a diagram; introduce variables with quantities that can change and constants that do not change. Denote DECREASE with a negative sign. B.) State the EQUATION, valid at any time t, among the variables involved.

3. C.) Implicitly differentiate the equation with respect to time in order to obtain a relationship about the RATES of the variables valid at any time t. D.) NOW, and NOT BEFORE, introduce specific value(s) which the variables take at the instant in the equation. E.) Answer the question and include proper units in the answer.

4. II. Examples A.) A 12 foot ladder stands against a vertical wall. If the lower end of the ladder slips away from the wall at a rate of 2 ft./sec., how fast is the top of the ladder coming down the wall when the ladder is 4 feet above the ground? 12 x y

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6. B.) A conical reservoir 12 feet deep and 12 feet across the top is being filled with water at a rate of 5 cubic feet per minute. How fast is the water level rising when the water is 4 feet deep? Diagram: 6 12 r h

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8. C.) A stone is dropped into a small pond. Concentric circular ripples spread out and the radius of the disturbed region increases at a rate of 16 cm/sec. At what rate does the area of the disturbed region increase when its radius is 4 cm? Diagram: r

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10. D.) Two roads meet at a right angle. Two cars leave the intersection at the same instant. One car travels directly north at a speed of 45 mph, the other travels directly west at a speed of 60 mph. After 25 minutes, how fast is the distance between them increasing? Diagram:

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12. E.) A spherical balloon is deflating so that its radius is decreasing at the of 2 in./sec. How fast is the volume of the balloon decreasing when the volume is 288π in. 3 ?

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14. F.) A Ferris wheel with a radius of 25 feet is revolving at the rate of 10 radians per minute. How fast is a passenger falling when the passenger is 15 feet higher than the center of the Ferris wheel and the passenger is on his way back down? Diagram:

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