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 Qnrepresent the quantity, in milligrams, of the drug in the body right after the nthdose. a.[2 points] Express Qnas a series. 12102525(0.03)25(0.03)...25(0.03)25(0.03)nniniQ−−==++++=∑. 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: 025252525(0.03)100100971(0.03)97100ii∞====⋅<−∑. 2.[8 points] Suppose that xmeasures 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 3if 01( )0otherwisecxxp x⎧≤≤=⎨⎩a.[4 points] Determine the value of cthat ensures that p(x) is a density function. 114300( )1111444xcp x dxcx dxcc∞−∞⎡⎤=⇒=⇒=⇒=⇒=⎢⎥⎣⎦∫∫.