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Section 3.5 Exponential and Logarithmic Models 245<br />
27. (a) ––CONTINUED––<br />
Turkey: 0, 65.7, 10, 73.3<br />
a 65.7 and 73.3 65.7e b10 ⇒ b 0.01095<br />
y 65.7e 0.01095t<br />
For 2030, y 91.2 million.<br />
(b) The constant b gives the growth rates.<br />
(c) The constant b is negative for South Africa.<br />
28.<br />
P 372.55e 0.01052t<br />
(a) Decreasing because the exponent is negative.<br />
(b) For 1990, t 0 and P 372,550 people.<br />
For 2000, t 10 and P 335,349 people.<br />
For 2004, t 14 and P 321,530 people.<br />
(c) 300 372.55e 0.01052t ⇒ t 20.6, or 2010<br />
29. (a)<br />
180 134.0e k10<br />
10k ln 180<br />
134.0<br />
k 0.0295<br />
(b) For 2010, t 20 and<br />
P 134.0e 0.029520 241,734 people.<br />
30.<br />
P 258.0e kt 31.<br />
(a)<br />
478 258.0e k10<br />
10k ln <br />
478<br />
258<br />
k 0.0617<br />
(b) For 2010, t 20 and<br />
P 258e 0.061720 886,215 people.<br />
y Ce kt<br />
1<br />
2 C Ce1599k<br />
ln <br />
1<br />
2 1599k<br />
k ln12<br />
1599<br />
When t 100, we have<br />
y Ce ln121001599 0.958C, or 95.8%.<br />
© Houghton Mifflin Company. All rights reserved.<br />
32.<br />
33. (a)<br />
y Ce kt<br />
1<br />
2 C Ce5715k<br />
ln <br />
1<br />
2 5715k<br />
(b)<br />
k ln12<br />
5715<br />
Vt 3394t 30,788<br />
V2 24,000 ⇒ 24,000 30,788e 2k<br />
V 30,788e 0.1245t<br />
⇒ m 3394<br />
The ancient charcoal has only 15% as much<br />
radioactive carbon.<br />
0.15C Ce ln125715t<br />
ln 0.15 ln12<br />
5715 t<br />
V ae kt , V0 30,788 ⇒ b 30,788<br />
t 15,642<br />
V mt b, V0 30,788 ⇒ b 30,788<br />
V2 24,000 ⇒ 24,000 2m 30,788<br />
⇒ k 1 2 ln 24,000<br />
30,788 0.1245<br />
years<br />
(c)<br />
31,000<br />
0 15<br />
0<br />
(d) The exponential model depreciates faster in the<br />
first year.<br />
(e) Answers will vary.
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