f09exam2solutions - Fall 2009 Math 113 Exam #2 Solutions 1....

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Fall 2009 Math 113 Exam #2 Solutions 1. An asteroid hits the Atlantic Ocean and creates an expanding circular wave. If the area enclosed by this wave increases at the rate of 200 km 2 /min, how fast is the diameter of the wave expanding when its radius is 20 km? Answer: We know that the area enclosed by the wave is given by A ( t ) = πr ( t ) 2 and that the diameter is 2 r ( t ). From this latter, the rate of change of the diameter (which is what we’re trying to determine) is 2 r 0 ( t ). Differentiating the expression for A , A 0 ( t ) = 2 πr ( t ) r 0 ( t ) . Therefore, r 0 ( t ) = A 0 ( t ) 2 πr ( t ) . At the time t 0 that we’re interested in, A 0 ( t 0 ) = 200 and r ( t 0 ) = 20. Plugging these values into the above equation, we see that r 0 ( t 0 ) = 200 2 π (20) = 200 40 π = 5 π . Therefore, at this instant the diameter is increasing at a rate of 2 r 0 ( t 0 ) = 2 5 π = 10 π km/min . 2. Suppose 10 bacteria are left to grow in a petri dish. At least for the first few hours, the rate of population growth of this bacterial culture will be proportional to the size of the culture. If there are
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This note was uploaded on 01/18/2012 for the course MATH 2250 taught by Professor Chestkofsky during the Fall '08 term at University of Georgia Athens.

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f09exam2solutions - Fall 2009 Math 113 Exam #2 Solutions 1....

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