Chapter 3, Problem 129RE

a.

To determine

To find : the numbers of weeks it took the class to read at average rates of $45$ words per minute

Answer to Problem 129RE

In $11.08weeks$ it took the class to read at average rates of $45$ words per minute

Explanation of Solution

Given information : The question is

The average number of words per minute that the students in a first grade class could read after $t$ weeks of school is modeled by $N=\frac{62}{1+5.4e^{-0.24t}}$

Calculation:

  $\begin{array}{l}N=\frac{62}{1+5.4e^{-0.24t}}\\ N=45\\ 45=\frac{62}{1+5.4e^{-0.24t}}\\ 45+243e^{-0.24t}=62\\ 243e^{-0.24t}=17\\ e^{-0.24t}=\frac{17}{243}\\ \ln e^{-0.24t}=\ln \frac{17}{243}\\ -0.24t=\ln \frac{17}{243}\\ t=\frac{\ln \frac{17}{243}}{-0.24}\\ t\approx 11.08weeks\end{array}$

b.

To determine

To find : the numbers of weeks it took the class to read at average rates of $61$ words per minute

Answer to Problem 129RE

In $24.16weeks$ it took the class to read at average rates of $61$ words per minute

Explanation of Solution

Given information : The question is

The average number of words per minute that the students in a first grade class could read after $t$ weeks of school is modeled by $N=\frac{62}{1+5.4e^{-0.24t}}$

Calculation:

  $\begin{array}{l}N=\frac{62}{1+5.4e^{-0.24t}}\\ N=61\\ 61=\frac{62}{1+5.4e^{-0.24t}}\\ 61+329.4e^{-0.24t}=62\\ 329.4e^{-0.24t}=1\\ e^{-0.24t}=\frac{1}{329.4}\\ \ln e^{-0.24t}=\ln \frac{1}{329.4}\\ -0.24t=\ln \frac{1}{329.4}\\ t=\frac{\ln \frac{1}{329.4}}{-0.24}\\ t\approx 24.16weeks\end{array}$