## Calculus (3rd Edition)

Published by W. H. Freeman

# Chapter 7 - Exponential Functions - 7.4 Exponential Growth and Decay - Exercises - Page 349: 5

#### Answer

$$N'(t)=\frac{\ln2}{3}N(t).$$ $$N(t)=P_0e^{kt}=e^{t\frac{\ln 2}{3}}.$$ $$N(60)= e^{20\ln 2}.$$

#### Work Step by Step

Since the doubling time is given by $\frac{\ln 2}{k}$, then we have $$3=\frac{\ln 2}{k}\Longrightarrow k=\frac{\ln2}{3}.$$ The differential equation is given by $$N'(t)=kN(t)\Longrightarrow N'(t)=\frac{\ln2}{3}N(t).$$ When $t=0$, we have $P_0=N(0)=1$, then $$N(t)=P_0e^{kt}=e^{t\frac{\ln 2}{3}}.$$ At $t=60$, we have $$N(60)= e^{60\frac{\ln 2}{3}}=e^{60\frac{\ln 2}{3}}=e^{20\ln 2}.$$

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