Chapter 8, Problem 5PS

a.

To determine

Find the number of subscribers in one year for each company.

Answer to Problem 5PS

The subscribers for the companies (in one year) are

To gold $\to$ $28,750$

To galaxy $\to$ $35,750$

To nonsubscriber $\to$ $35,500$

Explanation of Solution

Given:

The percent changes in satellite subscription each year matrix is given below.

  

The given subscribers for the companies are

To gold $\to$ $25,000$

To galaxy $\to$ $30,000$

To nonsubscriber $\to$ $45,000$

Calculation:

The current population in the three companies ( matrix form).

The percent changes matrix is

  $p=\left[\begin{array}{ccc}0.70& 0.15& 0.15\\ 0.20& 0.80& 0.15\\ 0.10& 0.05& 0.70\end{array}\right]$

The populations in three companies in one year $=ps$

  $\begin{array}{l}ps=\left[\begin{array}{ccc}0.70& 0.15& 0.15\\ 0.20& 0.80& 0.15\\ 0.10& 0.05& 0.70\end{array}\right]\left[\begin{array}{l}25,000\\ 30,000\\ 45,000\end{array}\right]\\ =\left[\begin{array}{c}0.70\times 25,000+0.15\times 30000+0.15\times 45000\\ 0.20\times 25,000+0.80\times 30000+0.15\times 45000\\ 0.10\times 25,000+0.05\times 30000+0.70\times 45000\end{array}\right]\\ =\left[\begin{array}{c}28,750\\ 35,750\\ 35,500\end{array}\right]\end{array}$

Hence in one year, the subscribers for the companies are

To gold $\to$ $28,750$

To galaxy $\to$ $35,750$

To nonsubscriber $\to$ $35,500$

b.

To determine

Find the number of subscribers in two year for each company.

Answer to Problem 5PS

The subscribers for the companies (in two years) are

To gold $\to$ $30812$

To galaxy $\to$ $39675$

To nonsubscriber $\to$ $29512$

Explanation of Solution

Given:

The percent changes in satellite subscription each year matrix is given below.

  

The given subscribers for the companies are

To gold $\to$ $25,000$

To galaxy $\to$ $30,000$

To nonsubscriber $\to$ $45,000$

Calculation:

The current population in the three companies ( matrix form).

The percent changes matrix is

  $p=\left[\begin{array}{ccc}0.70& 0.15& 0.15\\ 0.20& 0.80& 0.15\\ 0.10& 0.05& 0.70\end{array}\right]$

The populations in three companies in two year $=p^{2}s$

  $\begin{array}{l}{p}^{2}s={\left[\begin{array}{ccc}0.70& 0.15& 0.15\\ 0.20& 0.80& 0.15\\ 0.10& 0.05& 0.70\end{array}\right]}^2\left[\begin{array}{l}25,000\\ 30,000\\ 45,000\end{array}\right]\\ =\left[\begin{array}{ccc}\frac{107}{200}& \frac{93}{200}& \frac{93}{400}\\ \frac{63}{200}& \frac{271}{400}& \frac{51}{200}\\ \frac{3}{20}& \frac{9}{100}& \frac{41}{80}\end{array}\right]\left[\begin{array}{l}25,000\\ 30,000\\ 45,000\end{array}\right]\\ =\left[\begin{array}{c}30812\\ 39675\\ 29512\end{array}\right]\end{array}$

Hence in two year, the subscribers for the companies are

To gold $\to$ $30812$

To galaxy $\to$ $39675$

To nonsubscriber $\to$ $29512$

c.

To determine

Find the number of subscribers in two year for each company.

Answer to Problem 5PS

The subscribers for the companies (in three years) are

To gold $\to$ $31946$

To galaxy $\to$ $42329$

To nonsubscriber $\to$ $25723$

Explanation of Solution

Given:

The percent changes in satellite subscription each year matrix is given below.

  

The given subscribers for the companies are

To gold $\to$ $25,000$

To galaxy $\to$ $30,000$

To nonsubscriber $\to$ $45,000$

Calculation:

The current population in the three companies ( matrix form).

The percent changes matrix is

  $p=\left[\begin{array}{ccc}0.70& 0.15& 0.15\\ 0.20& 0.80& 0.15\\ 0.10& 0.05& 0.70\end{array}\right]$

The populations in three companies in three years $=p^{3}s$

  $\begin{array}{l}{p}^{3}s={\left[\begin{array}{ccc}0.70& 0.15& 0.15\\ 0.20& 0.80& 0.15\\ 0.10& 0.05& 0.70\end{array}\right]}^3\left[\begin{array}{l}25,000\\ 30,000\\ 45,000\end{array}\right]\\ =\left[\begin{array}{ccc}0.44425& 0.277875& 0.277875\\ 0.3815& 0.602& 0.327375\\ 0.17425& 0.120125& 0.39475\end{array}\right]\left[\begin{array}{l}25,000\\ 30,000\\ 45,000\end{array}\right]\\ =\left[\begin{array}{c}31946\\ 42329\\ 25723\end{array}\right]\end{array}$

Hence in three years, the subscribers for the companies are

To gold $\to$ $31946$

To galaxy $\to$ $42329$

To nonsubscriber $\to$ $25723$

d.

To determine

Explain the (subscriber and nonsubscriber) the results of one to three years.

Answer to Problem 5PS

The subscribers of the companies gold and galaxy are increasing and the number of nonsubscribers are decreasing.

Explanation of Solution

Given:

The percent changes in satellite subscription each year matrix is given below.

  

The given subscribers for the companies are

To gold $\to$ $25,000$

To galaxy $\to$ $30,000$

To nonsubscriber $\to$ $45,000$

Calculation:

The current population in the three companies ( matrix form).

The percent changes matrix is

The subscribers for the companies (in one year) are

To gold $\to$ $28,750$

To galaxy $\to$ $35,750$

To nonsubscriber $\to$ $35,500$

The subscribers for the companies (in two years) are

To gold $\to$ $30812$

To galaxy $\to$ $39675$

To nonsubscriber $\to$ $29512$

The subscribers for the companies (in three years) are

To gold $\to$ $31946$

To galaxy $\to$ $42329$

To nonsubscriber $\to$ $25723$

From the above results, the subscribers of the companies gold and galaxy are increasing and the number of nonsubscribers are decreasing.