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Exam1_Solutions_W11 - Math 115 First Midterm February 8...

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Math 115— First Midterm February 8, 2011 Name: EXAM SOLUTIONS Instructor: Section: 1. Do not open this exam until you are told to do so. 2. This exam has 10 pages including this cover. There are 9 problems. Note that the problems are not of equal difficulty, so you may want to skip over and return to a problem on which you are stuck. 3. Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out to your instructor when you hand in the exam. 4. Please read the instructions for each individual problem carefully. One of the skills being tested on this exam is your ability to interpret mathematical questions, so instructors will not answer questions about exam problems during the exam. 5. Show an appropriate amount of work (including appropriate explanation) for each problem, so that graders can see not only your answer but how you obtained it. Include units in your answer where that is appropriate. 6. You may use any calculator except a TI-92 (or other calculator with a full alphanumeric keypad). However, you must show work for any calculation which we have learned how to do in this course. You are also allowed two sides of a 3 ′′ × 5 ′′ note card. 7. If you use graphs or tables to find an answer, be sure to include an explanation and sketch of the graph, and to write out the entries of the table that you use. 8. Turn off all cell phones and pagers , and remove all headphones. 9. You must use the methods learned in this course to solve all problems. 10. Note that problems 6–9 will be graded giving very little partial credit. Problem Points Score 1 12 2 12 3 12 4 10 5 12 6 12 7 12 8 12 9 6 Total 100
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Math 115 / Exam 1 (February 8, 2011) page 2 1 . [12 points] Cyanide is used in solution to isolate elemental gold in gold mines. This unfortu- nately may result the groundwater near mines being contaminated with cyanide, which then must be removed. Suppose that at a certain mine site cyanide is removed from the groundwa- ter starting in 2005. The concentration, c (in ppm), of cyanide found in the groundwater at the site t years after the year 2005 is given in the following table. t (years) 0 1 2 c (ppm) 25.0 21.8 18.9 (Values for c are rounded.) a . [4 points] Find the equation of an exponential model for c ( t ). Show your work. Solution: Let c ( t ) = c 0 a t . Then c (0) = c 0 = 25 and c (1) = 25 a = 21 . 8, so a = 21 . 8 / 25 = 0 . 87. Thus c ( t ) = 25 (0 . 87) t . Note that c (2) = 25 (0 . 87) 2 = 18 . 9, the value given in the table. Alternately, let c ( t ) = c 0 e kt = 25 e kt . Then c (1) = 25 e k = 21 . 8, so that k = ln(21 . 8 / 25) = ln(0 . 87) = 0 . 14. Thus we may also write c ( t ) = 25 e 0 . 14 t . As before, this gives c (2) = 18 . 9. b . [4 points] Using your equation from (a), how many years will it take for the concentration of cyanide to be reduced to 10 ppm? Show all of your work. Solution: We want c ( t ) = 25 (0 . 87) t = 10, so that t = ln(10 / 25) / ln( . 87) = 6 . 6 years.
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