Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 1 - Review - Exercises - Page 70: 28

Answer

(a) On the graph, we can see that the population reaches 900 after approximately 4.5 years. (b) This function tells us how much time is required (in years) for a population to reach a level of $P$. (c) It takes 4.4 years for the population to reach 900.
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Work Step by Step

(a) On the graph, we can see that the population reaches 900 after approximately 4.5 years. (b) We can find the inverse: $P = \frac{100,000}{100+900e^{-t}}$ $100+900e^{-t} = \frac{100,000}{P}$ $900e^{-t} = \frac{100,000}{P}-100$ $e^{-t} = \frac{1}{900}\cdot(\frac{100,000}{P}-100)$ $-t = ln[\frac{1}{900}\cdot(\frac{100,000}{P}-100)]$ $t = -ln[\frac{1}{900}\cdot(\frac{100,000}{P}-100)]$ This function tells us how much time is required (in years) for a population to reach a level of $P$. (c) We can find the time required for the population to reach 900: $t = -ln[\frac{1}{900}\cdot(\frac{100,000}{P}-100)]$ $t = -ln[\frac{1}{900}\cdot(\frac{100,000}{900}-100)]$ $t = -ln(0.012345679)$ $t = 4.4$ It takes 4.4 years for the population to reach 900.
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