Chapter 3.4, Problem 57E

(a)

To determine

To find: the initial number of students who have heard the rumor.

Answer to Problem 57E

The initial number of students who have heard the rumor is $18$ .

Explanation of Solution

Given information :

The spread of a rumor is modeled by the equation $P\left(t\right)=\frac{300}{1+2^{4-t}}$ where $t$ days.

From initial number, plug $t=0$ in the equation $P\left(t\right)=\frac{300}{1+2^{4-t}}$ .

  $\begin{array}{c}P\left(0\right)=\frac{300}{1+2^{4-0}}\\ =\frac{300}{1+2^4}\\ =\frac{300}{1+16}\\ =\frac{300}{17}\\ \approx 18\end{array}$

The initial number of students who have heard the rumor is $18$ .

(b)

To determine

To find: the rate at which rumor spreading after 4 days.

Answer to Problem 57E

The rate at which rumor spreading after 4 days is $52$ students per day.

Explanation of Solution

Given information :

The spread of a rumor is modeled by the equation $P\left(t\right)=\frac{300}{1+2^{4-t}}$ where $t$ days.

Formula needed :

  $\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$ and $\frac{d}{dx}\left(a^x\right)=a^x\ln a$

Differentiate $P\left(t\right)=\frac{300}{1+2^{4-t}}$ with respect to $t$ .

  $\begin{array}{c}{P}^{\prime }\left(t\right)=300\cdot \frac{d}{dt}{\left(1+2^{4-t}\right)}^{-1}\\ =300\cdot \left[-{\left(1+2^{4-t}\right)}^{-2}\right]\cdot \frac{d}{dt}\left(1+2^{4-t}\right)\\ =300\cdot \left[-{\left(1+2^{4-t}\right)}^{-2}\right]\cdot 2^{4-t}\ln 2\frac{d}{dt}\left(4-t\right)\\ =300\cdot \left[-{\left(1+2^{4-t}\right)}^{-2}\right]\cdot 2^{4-t}\ln 2\left(-1\right)\end{array}$

Further simplifying,

  $P^{\prime }\left(t\right)=\frac{300\cdot \ln 2}{{\left(1+2^{4-t}\right)}^2}{2}^{4-t}$

Substitute $t=4$ in the above function.

  $\begin{array}{c}{P}^{\prime }\left(4\right)=\frac{300\cdot \ln 2}{{\left(1+2^{4-4}\right)}^2}{2}^{4-4}\\ =\frac{300\cdot \ln 2}{{\left(1+2^0\right)}^2}{2}^0\\ =75\ln 2\\ \approx 52\end{array}$

The rate at which rumor spreading after 4 days is $52$ students per day.

(c)

To determine

To find: the maximum rate and time at which rumor spread.

Answer to Problem 57E

The maximum rate and time at which rumor spreading is $52$ students per day and 4 days respectively.

Explanation of Solution

Given information :

The spread of a rumor is modeled by the equation $P\left(t\right)=\frac{300}{1+2^{4-t}}$ where $t$ days.

Formula needed :

  $\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$ and $\frac{d}{dx}\left(a^x\right)=a^x\ln a$

Differentiate $P^{\prime }\left(t\right)=\frac{300\cdot \ln 2}{{\left(1+2^{4-t}\right)}^2}{2}^{4-t}$ with respect to $t$ .

  $\begin{array}{c}{P}^{″}\left(t\right)=300\cdot \ln 2\cdot \frac{d}{dt}\left(2^{4-t}\cdot {\left(1+2^{4-t}\right)}^{-2}\right)\\ =300\cdot \ln 2\cdot \left[2^{4-t}\cdot \frac{d}{dt}{\left(1+2^{4-t}\right)}^{-2}+{\left(1+2^{4-t}\right)}^{-2}\cdot \frac{d}{dt}\left(2^{4-t}\right)\right]\\ =300\cdot \ln 2\cdot \left[2^{4-t}\cdot 2{\left(1+2^{4-t}\right)}^{-3}{2}^{4-t}\ln 2+{\left(1+2^{4-t}\right)}^{-2}\cdot \left(-2^{4-t}\ln 2\right)\right]\\ =300\cdot \ln 2\cdot \left(2^{4-t}\ln 2\right){\left(1+2^{4-t}\right)}^{-2}\left[2{\left(1+2^{4-t}\right)}^{-1}{2}^{4-t}-1\right]\end{array}$

Take $P^{″}\left(t\right)=0$ .

  $\begin{array}{c}300\cdot \ln 2\cdot \left(2^{4-t}\ln 2\right){\left(1+2^{4-t}\right)}^{-2}\left[2{\left(1+2^{4-t}\right)}^{-1}{2}^{4-t}-1\right]=0\\ \frac{2^{4-t}}{{\left(1+2^{4-t}\right)}^2}\left[\frac{2\cdot 2^{4-t}}{\left(1+2^{4-t}\right)}-1\right]=0\left[300\cdot {\left(\ln 2\right)}^2\ne 0\right]\\ \frac{2^{4-t}}{{\left(1+2^{4-t}\right)}^2}\left[\frac{2\cdot 2^{4-t}-1-2^{4-t}}{\left(1+2^{4-t}\right)}\right]=0\\ \frac{2^{4-t}}{{\left(1+2^{4-t}\right)}^2}\left[\frac{-1+2^{4-t}}{\left(1+2^{4-t}\right)}\right]=0\end{array}$

After solving and simplifying,

  $t=4$ days

Substitute $t=4$ in the function $P^{\prime }\left(t\right)=\frac{300\cdot \ln 2}{{\left(1+2^{4-t}\right)}^2}{2}^{4-t}$ .

  $\begin{array}{c}{P}^{\prime }\left(4\right)=\frac{300\cdot \ln 2}{{\left(1+2^{4-4}\right)}^2}{2}^{4-4}\\ =\frac{300\cdot \ln 2}{{\left(1+2^0\right)}^2}{2}^0\\ =75\ln 2\\ \approx 52\end{array}$

The maximum rate and time at which rumor spreading is $52$ students per day and 4 days respectively.