Calculus: Early Transcendentals 8th Edition

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

Chapter 3 - Section 3.4 - The Chain Rule - 3.4 Exercises - Page 206: 84

Answer

(a) $\lim\limits_{t \to \infty}p(t) = 1$ (b) $\frac{dp}{dt} = \frac{kae^{-kt}}{(1+ae^{-kt})^2}$ (c) It will take about 7.4 hours for 80% of the population to hear the rumor.
1557405218

Work Step by Step

(a) $p(t) = \frac{1}{1+ae^{-kt}}$ $\lim\limits_{t \to \infty}p(t) = \lim\limits_{t \to \infty}\frac{1}{1+ae^{-kt}}$ $\lim\limits_{t \to \infty}p(t) = \lim\limits_{t \to \infty}\frac{1}{1+a(0)}$ $\lim\limits_{t \to \infty}p(t) = 1$ (b) $\frac{dp}{dt} = \frac{0-(1)(-kae^{-kt})}{(1+ae^{-kt})^2} = \frac{kae^{-kt}}{(1+ae^{-kt})^2}$ (c) $p(t) = \frac{1}{1+10e^{-0.5t}}$ On the graph, we can see that it will take about 7.4 hours for 80% of the population to hear the rumor.
Small 1557405218
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.