Chapter 1, Problem 30CR

(a)

To determine

To graph: the approximate number of students (in millions) enrolled in public schools of U.S from 2009 to 2016.

Explanation of Solution

Given information: The mathematical model for the number of students (in millions) enrolled in public schools of U.S from 2009 to 2016 is:

  $n\left(t\right)=\{\begin{array}{l}-0.35t^2+7.7t+26,9\le t\le 12\\ -0.187t^3+8.22t^2-120.4t+654,13\le t\le 16\end{array}$

Where t is the year, with $t=9$ corresponding to the year 2009.

Calculation:

The table for the approximate number of students (in millions) enrolled is given below:

Year (t )	Number of students $n\left(t\right)$
2009	66.95
2010	68
2011	68.35
2012	68
2013	67.141
2014	66.392
2015	66.375
2016	65.968

The above data can be graphically shown as:

  

(b)

To determine

whether the piecewise defined function could be used to predict the enrollments for years outside the domain.

Explanation of Solution

Given information: The mathematical model for the number of students (in millions) enrolled in public schools of U.S from 2009 to 2016 is:

  $n\left(t\right)=\{\begin{array}{l}-0.35t^2+7.7t+26,9\le t\le 12\\ -0.187t^3+8.22t^2-120.4t+654,13\le t\le 16\end{array}$

Where t is the year, with $t=9$ corresponding to the year 2009.

The given mathematical model is piecewise defined for the years 2009 to 2012 and 2013 to 2016.It can be seen from the graph that number of students enrolled in these years comes under the range 60 and 70(in millions).

If there is no change in the rate of number of students enrolled in the years outside the domain, then the given mathematical model can be used for $t\le 8$ and $t\ge 17$ .