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Section 3.5 Exponential and Logarithmic Models 245 27. (a) ––CONTINUED–– Turkey: 0, 65.7, 10, 73.3 a 65.7 and 73.3 65.7e b10 ⇒ b 0.01095 y 65.7e 0.01095t For 2030, y 91.2 million. (b) The constant b gives the growth rates. (c) The constant b is negative for South Africa. 28. P 372.55e 0.01052t (a) Decreasing because the exponent is negative. (b) For 1990, t 0 and P 372,550 people. For 2000, t 10 and P 335,349 people. For 2004, t 14 and P 321,530 people. (c) 300 372.55e 0.01052t ⇒ t 20.6, or 2010 29. (a) 180 134.0e k10 10k ln 180 134.0 k 0.0295 (b) For 2010, t 20 and P 134.0e 0.029520 241,734 people. 30. P 258.0e kt 31. (a) 478 258.0e k10 10k ln 478 258 k 0.0617 (b) For 2010, t 20 and P 258e 0.061720 886,215 people. y Ce kt 1 2 C Ce1599k ln 1 2 1599k k ln12 1599 When t 100, we have y Ce ln121001599 0.958C, or 95.8%. © Houghton Mifflin Company. All rights reserved. 32. 33. (a) y Ce kt 1 2 C Ce5715k ln 1 2 5715k (b) k ln12 5715 Vt 3394t 30,788 V2 24,000 ⇒ 24,000 30,788e 2k V 30,788e 0.1245t ⇒ m 3394 The ancient charcoal has only 15% as much radioactive carbon. 0.15C Ce ln125715t ln 0.15 ln12 5715 t V ae kt , V0 30,788 ⇒ b 30,788 t 15,642 V mt b, V0 30,788 ⇒ b 30,788 V2 24,000 ⇒ 24,000 2m 30,788 ⇒ k 1 2 ln 24,000 30,788 0.1245 years (c) 31,000 0 15 0 (d) The exponential model depreciates faster in the first year. (e) Answers will vary.
246 Chapter 3 Exponential and Logarithmic Functions 34. Let t 0 correspond to 2005. (a) (b) V mt b, V0 1150 ⇒ b 1150 V2 550 ⇒ 550 m2 1150 ⇒ m 300 Vt 300t 1150 V ae kt , V0 1150 ⇒ a 1150 V2 550 ⇒ 550 1150e 2k ⇒ k 1 2 Vt 1150e 0.3688t ln 55 115 0.3688 (c) 1200 0 10 0 (d) The exponential model depreciates faster in the first year. (e) Answers will vary. 35. St 1001 e kt 36. (a) (b) (c) 15 1001 e k1 85 100e k St 1001 e 0.1625t 100 0 0 k ln 0.85 k 0.1625 50 S5 1001 e 0.16255 55.625 55,625 units S 101 e kx x 5 in hundreds S 2.5 in thousands (a) 2.5 101 e k5 0.25 1 e 5k e 5k 0.75 5k ln 0.75 k 0.0575 S 101 e 0.0575x (b) When x 7, S 101 e 0.05757 3.314 which corresponds to 3314 units. 37. y 0.0266e x1002 450 , 70 ≤ x ≤ 115 38. y 0.7979e x5.42 0.5 (a) 0.05 (a) 1 70 0 (b) Maximum point is x 100, the average IQ score. 115 4 0 (b) About 5.4 hours 7 © Houghton Mifflin Company. All rights reserved.
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