Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 4 - Section 4.6 - Modeling with Exponential Functions - 4.6 Exercises - Page 378: 2

Answer

a. $n(t)=25\cdot 2^{t/5}$ b. about 303 bacteria c. after about $76.4$ hours

Work Step by Step

a. Given: n(0)=25 and n(5)=50 (double after 5 hours) $50=25\cdot 2^{5/a}$ ... solve for a, start by dividing both sides with $25$... $ 2=2^{5/a}\qquad$ ...apply $\log_{2}$ to both sides $1=\displaystyle \frac{5}{a}\qquad /\times a$ $a=5$ The model is $n(t)=25\cdot 2^{t/5}$ b. t=$18$, $ n(t)=25\cdot 2^{18/5}\approx$303.143313302 about 303 bacteria c. Solve for t when n(t)=1,000,000 $1,000,000=25\cdot 2^{t/5}\qquad /\div 25$ $ 40,000=2^{t/5}\qquad$ ...apply $\log()$ to both sides $\displaystyle \log 40,000=\frac{t}{5}\log 2\qquad/\times\frac{5}{\log 5}$ $ t=\displaystyle \frac{5\log 40,000}{\log 2}\approx$76.4385618977 after about $76.4$ hours
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