finalsol_v1

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 21 0 25 25(0.03) 25(0.03) ... 25(0.03) 25 (0.03) n ni 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 () 0o t h e r w i s e cx x px = 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 44 xc p x dx cx dx c c −∞ ⎡⎤ =⇒ =⇒ =⇒ = ⎢⎥ ⎣⎦ ∫∫ .
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b. [4 points] What is the mean time for the students to complete the exam? 1 11 5 34 00 0 44 () 4 55 x xp x dx xcx dx x dx −∞ ⎡⎤ ⇒⇒ = ⎢⎥ ⎣⎦ ∫∫ hours. 3. [ 8 points] Suppose f ( x , y ) is a linear function with the values a. [4 points] Write the equation of f in the form f ( x , y ) = mx + ny + c . m = -3, n = 2, c = 4 (where x and y are both 0), so we have that: f ( x , y ) = -3 x + 2 y + 4.
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finalsol_v1 - Math 10C Spring 2007 Final Exam Solutions...

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