finalsol_v1 - Math 10C Spring 2007 Final Exam Solutions...

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Math 10C Spring 2007 Final Exam Solutions – Version 1 1. [6 points] Suppose a drug is taken in 25mg doses four times a day (i.e. every six hours). It is known that at the end of the six hours, 3% of the drug remains in the body. Let Q n represent the quantity, in milligrams, of the drug in the body right after the n th dose. a. [2 points] Express Q n as a series. 1 2 1 0 25 25(0.03) 25(0.03) ... 25(0.03) 25 (0.03) n n i n i Q = = + + + + = . b. [2 points] Explain why the series from part (a) is a geometric series. We have a constant ratio, 0.03, between successive terms. c. [2 points] Assuming that the doses of the drug continue indefinitely, will the amount of the drug in the body ever exceed 100mg? Explain. No, since the infinite sum is equal to: 0 25 25 25 25 (0.03) 100 100 97 1 (0.03) 97 100 i i = = = = < . 2. [8 points] Suppose that x measures the time it takes for a student to complete an exam for which the maximum time allowed is one hour. The distribution of the completion times by the students is given by the probability density function 3 if 0 1 ( ) 0 otherwise cx x p x = a. [4 points] Determine the value of c that ensures that p ( x ) is a density function. 1 1 4 3 0 0 ( ) 1 1 1 1 4 4 4 x c p x dx cx dx c c −∞ = = = = = .