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Section 3.4 Solving Exponential and Logarithmic Equations 237 132. (a) (b) 2000 1000e 0.06t 133. (a) 2 e 0.06t ln 2 0.06t t ln 2 11.55 years 0.06 3000 1000e 0.06t 3 e 0.06t ln 3 0.06t t ln 3 18.31 years 0.06 (b) 2000 1000e 0.025t 2 e 0.025t ln 2 0.025t t ln 2 27.73 years 0.025 3000 1000e 0.025t 3 e 0.025t ln 3 0.025t t ln 3 43.94 years 0.025 134. (a) 136. (b) 2000 1000e 0.0375t 135. 2 e 0.0375t ln 2 0.0375t t ln 2 18.48 years 0.0375 3000 1000e 0.0375t 3 e 0.0375t ln 3 0.0375t p 5000 1 t ln 3 29.30 years 0.0375 4 p 500 0.5e 0.004x (a) (b) p 350 350 500 0.5e 0.004x 300 e 0.004x 0.004x ln 300 x 1426 units p 300 300 500 0.5e 0.004x 400 e 0.004x 0.004x ln 400 x 1498 units © Houghton Mifflin Company. All rights reserved. 0.002x 4 e (a) When p $600: (b) When p $400: 600 5000 1 0.12 1 4 0.88 0.002x 4 e 4 3.52 0.88e 0.002x 0.48 0.88e 0.002x 6 11 e0.002x ln 6 ln e0.002x 11 ln 6 11 0.002x x ln611 0.002 4 4 e 0.002x 4 4 e 0.002x 303 units 400 5000 1 0.08 1 4 0.92 4 e0.002x 4 3.68 0.92e 0.002x 0.32 0.92e 0.002x 8 23 e0.002x ln 8 ln e0.002x 23 x ln823 0.002 4 4 e 0.002x 4 4 e 0.002x 528 units
238 Chapter 3 Exponential and Logarithmic Functions 137. 7247 596.5 ln t 5800 596.5 ln t 1447 ln t 2.4258 t 11.3, or 2001 138. V 6.7e 48.1t , t > 0 (a) 10 (c) 1.3 6.7e 48.1t 1.3 6.7 e48.1t 0 0 1500 (b) As t → , V → 6.7. Horizontal asymptote: y 6.7 The yield will approach 6.7 million cubic feet per acre. ln 13 67 48.1 t t 48.1 29.3 years ln 1367 139. (a) 110 140. P 0.83 1 e 0.2n f m (a) 1 0 0 110 (b) From the graph we see horizontal asymptotes at y 0 and y 100. These represent the lower and upper percent bounds. (c) Males: 1 e 0.6114x69.71 2 0.6114x 69.71 ln 1 0.6114x 69.71 0 Females: 50 e 0.6114x69.71 1 50 1 e 0.66607x64.51 2 e 0.66607x64.51 1 0.66607x 64.51 ln 1 0.66607x 64.51 0 100 1 e 0.6114x69.71 x 69.71 inches 100 1 e 0.66607x64.51 x 64.51 inches −40 (b) Horizontal asymptotes: y 0, y 0.83 The upper asymptote, y 0.83, indicates that the proportion of correct responses will approach 0.83 as the number of trials increases. (c) When P 60% or P 0.60: 0.60 0.83 1 e 0.2n 1 e 0.2n 0.83 0.60 0 40 e 0.2n 0.83 0.60 1 ln e 0.2n ln 0.83 0.60 1 0.2n ln 0.83 0.60 1 0.83 ln 0.60 1 n 5 trials 0.2 © Houghton Mifflin Company. All rights reserved.
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238 Chapter 3 Exponential and Logarithmic Functions<br />
137.<br />
7247 596.5 ln t 5800<br />
596.5 ln t 1447<br />
ln t 2.4258<br />
t 11.3, or 2001<br />
138.<br />
V 6.7e 48.1t , t > 0<br />
(a)<br />
10<br />
(c)<br />
1.3 6.7e 48.1t<br />
1.3<br />
6.7 e48.1t<br />
0<br />
0<br />
1500<br />
(b) As t → , V → 6.7.<br />
Horizontal asymptote: y 6.7<br />
The yield will approach<br />
6.7 million cubic feet per acre.<br />
ln 13<br />
67 48.1<br />
t<br />
t 48.1 29.3 years<br />
ln 1367<br />
139. (a)<br />
110<br />
140.<br />
P 0.83<br />
1 e 0.2n<br />
f<br />
m<br />
(a)<br />
1<br />
0<br />
0<br />
110<br />
(b) From the graph we see horizontal asymptotes at<br />
y 0 and y 100. These represent the lower and<br />
upper percent bounds.<br />
(c) Males:<br />
1 e 0.6114x69.71 2<br />
0.6114x 69.71 ln 1<br />
0.6114x 69.71 0<br />
Females:<br />
50 <br />
e 0.6114x69.71 1<br />
50 <br />
1 e 0.66607x64.51 2<br />
e 0.66607x64.51 1<br />
0.66607x 64.51 ln 1<br />
0.66607x 64.51 0<br />
100<br />
1 e 0.6114x69.71<br />
x 69.71 inches<br />
100<br />
1 e 0.66607x64.51<br />
x 64.51 inches<br />
−40<br />
(b) Horizontal asymptotes: y 0, y 0.83<br />
The upper asymptote, y 0.83, indicates<br />
that the proportion of correct responses will<br />
approach 0.83 as the number of trials<br />
increases.<br />
(c) When P 60% or P 0.60:<br />
0.60 0.83<br />
1 e 0.2n<br />
1 e 0.2n 0.83<br />
0.60<br />
0<br />
40<br />
e 0.2n 0.83<br />
0.60 1<br />
ln e 0.2n ln <br />
0.83<br />
0.60 1 <br />
0.2n ln <br />
0.83<br />
0.60 1 <br />
0.83<br />
ln 0.60 1 <br />
n <br />
5 trials<br />
0.2<br />
© Houghton Mifflin Company. All rights reserved.
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