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Exam1_Solutions_W11

# Exam1_Solutions_W11 - Math 115 First Midterm February 8...

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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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