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Unformatted text preview: AMS 151.01 Applied Calculus I Spring 2008, Pratice Final Exam Solution Sketch 1. A population of animals oscillates sinusoidally between a low of 300 on January 1st, 2006 and a high of 1000 on July 1st, 2006. (a) Graph the population against time. (b) Use sine as your base function, find a formula for the population P as a function of time, t , measured in months since the start of the year. Solution: (b) Let P = A sin( Bt + C ) + D . The amplitude A can be computed by A = max min 2 = 1000 300 2 = 350 . B is determined by the period of the function according to the following equation B = 2 period Since period equal to 12 in this example, B = 2 / 12 = / 6. Then vertical shift D obtained by D = max A = 1000 350 = 650 . In order to find C , the phase shift, we use the fact that function reaches its minimum on January 1st, i.e. when t = 0. Thus sin ( C ) = 1. We can pick any value of C as long as it satisfies this equation, for example, C = / 2 ,C = 3 / 2. 2. In the early 1960s, radioactive strontium90 was released during atmospheric testing of nuclear weapons and got into the bones of people alive at the time. If the halflife of strontium90 is 40 years, (a) Write a formula for the quantity of strontium90, Q , left after t years, if the initial quantity is Q . (b) If the initial quantity of strontium90 was measured in 1960s, what percentage of the original amount will be left in 2006? Solution: (a) Let Q ( t ) = Q a t . a is the only unknown parameter. Because the halflife is 40 years, we have a 40 = 1 2 which yields a = 1 2 1 / 40 ....
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 Spring '08
 Zhang

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