Chapter 7, Problem 67RE

a.

To determine

To find what proportion of the townspeople have heard the rumor when it is spreading the fastest.

Answer to Problem 67RE

½ of the population of the townspeople have heard the rumor when it is spreading the fastest.

Explanation of Solution

Given information:

The spread of a rumor through a small town is modeled by $dy/dt=1.2y\left(1-y\right)$ .

Calculation:

In question its only asks for they people that will hear the rumor

i.e ½ of the population.

Hence, ½ of the population of the townspeople have heard the rumor when it is spreading the fastest.

b.

To determine

To find y explicitly as a function of t.

Answer to Problem 67RE

The required solution is, $y=\frac{1}{1+9e^{-1.2t}}$ .

Explanation of Solution

Given information:

The spread of a rumor through a small town is modeled by $dy/dt=1.2y\left(1-y\right)$ .

Calculation: The required solution is obtained as,

  $\begin{array}{l}\frac{dy}{dt}=1.2y\left(1-y\right)\frac{dy}{y\left(1-y\right)}=1.2t\\ {\displaystyle \int \left(\frac{1}{y}-\frac{1}{y-1}\right)dy=1.2t+C}\\ \ln \left|y\right|-\ln \left|y-1\right|=1.2t+C\\ -\ln \left|y\right|+\ln \left|y-1\right|=-1.2t+C\\ \ln \left|\frac{y-1}{y}\right|=C-1.2t\\ 1-\frac{1}{y}=C{e}^{-1.2t}\\ \frac{1}{y}=1+C{e}^{-1.2t}\\ y=\frac{1}{1+C{e}^{-1.2t}}\\ 0.1=\frac{1}{1+C}\\ C=9\end{array}$

Substituting C=9 in the equation $y=\frac{1}{1+C{e}^{-1.2t}}$ .

  $y=\frac{1}{1+9e^{-1.2t}}$

Hence, the required solution is, $y=\frac{1}{1+9e^{-1.2t}}$ .

c.

To determine

To find at what time t is the rumor spreading the fastest.

Answer to Problem 67RE

After 1.83 days it is introduced, the rumor spreading the fastest.

Explanation of Solution

Given information:

The spread of a rumor through a small town is modeled by $dy/dt=1.2y\left(1-y\right)$ .

Calculation: The rumor is spreading the fastest when half the townspeople know it,

  $\begin{array}{l}0.5=\frac{1}{1+9e^{-1.2t}}\\ 1+9e^{-1.2t}=2\\ t=1.83\end{array}$

Hence, after 1.83 days it is introduced, the rumor spreading the fastest.