Chapter 11.4, Problem 86AYU

  $\left(a\right)$

To determine

To calculate: The matrices that represent the sales data for city and suburbs for month of January and February.

Answer to Problem 86AYU

The matrices that represent the sales data for city and suburbs for month of January and February are $\begin{array}{l}\begin{array}{cccccc}& & \text{subcompact}& \text{intermediate}& & \text{SUV}\end{array}\\ \begin{array}{c}\text{city}\\ \text{suburbs}\end{array}\left[\begin{array}{cccccccccc}400& & & & 250& & & & 50& \\ 450& & & & 200& & & & 140& \end{array}\right]\\ \end{array}$ and

  $\begin{array}{l}\begin{array}{cccccc}& & \text{subcompact}& \text{intermediate}& & \text{SUV}\end{array}\\ \begin{array}{c}\text{city}\\ \text{suburbs}\end{array}\left[\begin{array}{cccccccccc}350& & & & 100& & & & 30& \\ 350& & & & 300& & & & 100& \end{array}\right]\\ \end{array}$ respectively.

Explanation of Solution

Given information:

The Rizza Ford sold 400 subcompact, 250 intermediate-size cars and 50 SUV’s in the city location and 450 subcompact, 200 intermediate-size cars and 140 SUV’s in suburbs for the month of January.

The Rizza Ford sold 350 subcompact, 100 intermediate-size cars and 30 SUV’s in the city location and 350 subcompact, 300 intermediate-size cars and 100 SUV’s in suburbs for the month of February.

Calculation:

Consider the provided information that the Rizza Ford sold 400 subcompact, 250 intermediate-size cars and 50 SUV’s in the city location and 450 subcompact, 200 intermediate-size cars and 140 SUV’s in suburbs for the month of January.

The Rizza Ford sold 350 subcompact, 100 intermediate-size cars and 30 SUV’s in the city location and 350 subcompact, 300 intermediate-size cars and 100 SUV’s in suburbs for the month of February.

The above data can be represented as a $2\times 3$ where each row represent the number of type of car sold in city and in suburbs.

The representation for sales for month of January is,

  $\begin{array}{l}\begin{array}{cccccc}& & \text{subcompact}& \text{intermediate}& & \text{SUV}\end{array}\\ \begin{array}{c}\text{city}\\ \text{suburbs}\end{array}\left[\begin{array}{cccccccccc}400& & & & 250& & & & 50& \\ 450& & & & 200& & & & 140& \end{array}\right]\\ \end{array}$

The representation for sales for month of February is,

  $\begin{array}{l}\begin{array}{cccccc}& & \text{subcompact}& \text{intermediate}& & \text{SUV}\end{array}\\ \begin{array}{c}\text{city}\\ \text{suburbs}\end{array}\left[\begin{array}{cccccccccc}350& & & & 100& & & & 30& \\ 350& & & & 300& & & & 100& \end{array}\right]\\ \end{array}$

  $\left(b\right)$

To determine

To calculate: The total sale done in January and February.

Answer to Problem 86AYU

The total sale done in January and February is $\begin{array}{l}\begin{array}{cccccc}& & \text{subcompact}& \text{intermediate}& & \text{SUV}\end{array}\\ \begin{array}{c}\text{city}\\ \text{suburbs}\end{array}\left[\begin{array}{cccccccccc}750& & & & 350& & & & 80& \\ 800& & & & 500& & & & 240& \end{array}\right]\\ \end{array}$ .

Explanation of Solution

Given information:

The representation for sales for month of January is,

  $\begin{array}{l}\begin{array}{cccccc}& & \text{subcompact}& \text{intermediate}& & \text{SUV}\end{array}\\ \begin{array}{c}\text{city}\\ \text{suburbs}\end{array}\left[\begin{array}{cccccccccc}400& & & & 250& & & & 50& \\ 450& & & & 200& & & & 140& \end{array}\right]\\ \end{array}$

The representation for sales for month of February is,

  $\begin{array}{l}\begin{array}{cccccc}& & \text{subcompact}& \text{intermediate}& & \text{SUV}\end{array}\\ \begin{array}{c}\text{city}\\ \text{suburbs}\end{array}\left[\begin{array}{cccccccccc}350& & & & 100& & & & 30& \\ 350& & & & 300& & & & 100& \end{array}\right]\\ \end{array}$

Calculation:

Consider the representation for sales for month of January is,

  $\begin{array}{l}\begin{array}{cccccc}& & \text{subcompact}& \text{intermediate}& & \text{SUV}\end{array}\\ \begin{array}{c}\text{city}\\ \text{suburbs}\end{array}\left[\begin{array}{cccccccccc}400& & & & 250& & & & 50& \\ 450& & & & 200& & & & 140& \end{array}\right]\\ \end{array}$

The representation for sales for month of February is,

  $\begin{array}{l}\begin{array}{cccccc}& & \text{subcompact}& \text{intermediate}& & \text{SUV}\end{array}\\ \begin{array}{c}\text{city}\\ \text{suburbs}\end{array}\left[\begin{array}{cccccccccc}350& & & & 100& & & & 30& \\ 350& & & & 300& & & & 100& \end{array}\right]\\ \end{array}$

Therefore, total sales done in month of January and February is the sum of the above matrices ,

  $\begin{array}{l}\begin{array}{cccccc}& & \text{subcompact}& \text{intermediate}& & \text{SUV}\end{array}\\ \begin{array}{c}\text{city}\\ \text{suburbs}\end{array}\left[\begin{array}{cccccccccc}400& & & & 250& & & & 50& \\ 450& & & & 200& & & & 140& \end{array}\right]+\\ \begin{array}{cccccc}& & \text{subcompact}& \text{intermediate}& & \text{SUV}\end{array}\\ \begin{array}{c}\text{city}\\ \text{suburbs}\end{array}\left[\begin{array}{cccccccccc}350& & & & 100& & & & 30& \\ 350& & & & 300& & & & 100& \end{array}\right]\\ =\begin{array}{cccccc}& & \text{subcompact}& \text{intermediate}& & \text{SUV}\end{array}\\ \begin{array}{c}\text{city}\\ \text{suburbs}\end{array}\left[\begin{array}{cccccccccc}750& & & & 350& & & & 80& \\ 800& & & & 500& & & & 240& \end{array}\right]\end{array}$

Therefore, total sale done in January and February is $\begin{array}{l}\begin{array}{cccccc}& & \text{subcompact}& \text{intermediate}& & \text{SUV}\end{array}\\ \begin{array}{c}\text{city}\\ \text{suburbs}\end{array}\left[\begin{array}{cccccccccc}750& & & & 350& & & & 80& \\ 800& & & & 500& & & & 240& \end{array}\right]\\ \end{array}$ .

  $\left(c\right)$

To determine

To calculate: The matrix representation of profit on each type of car.

Answer to Problem 86AYU

The matrix representation of profit on each type of car is $\left[\begin{array}{c}100\\ 150\\ 200\end{array}\right]\begin{array}{c}\text{subcompact}\\ \text{intermediate}\\ \text{SUV}\end{array}$ .

Explanation of Solution

Given information:

The profit on each subcompact car is $\text{\$100}$ , $\text{\$150}$ per intermediate and $\text{\$200}$ per SUV.

It is provided that the profit on each subcompact car is $\text{\$100}$ , $\text{\$150}$ per intermediate and $\text{\$200}$ per SUV.

Construct a 3 by 1 column matrix to represent the profit on each type of car.

Therefore, profit is

  $\left[\begin{array}{c}100\\ 150\\ 200\end{array}\right]\begin{array}{c}\text{subcompact}\\ \text{intermediate}\\ \text{SUV}\end{array}$

  $\left(d\right)$

To determine

To calculate: The profit of the company at each location.

Answer to Problem 86AYU

The profit of the company at each location is $\$346500$ .

Explanation of Solution

Given information:

The matrix representation of profit on each type of car is $\left[\begin{array}{c}100\\ 150\\ 200\end{array}\right]\begin{array}{c}\text{subcompact}\\ \text{intermediate}\\ \text{SUV}\end{array}$ .

The total sale done in January and February is $\begin{array}{l}\begin{array}{cccccc}& & \text{subcompact}& \text{intermediate}& & \text{SUV}\end{array}\\ \begin{array}{c}\text{city}\\ \text{suburbs}\end{array}\left[\begin{array}{cccccccccc}750& & & & 350& & & & 80& \\ 800& & & & 500& & & & 240& \end{array}\right]\\ \end{array}$ .

The Rizza Ford sold 400 subcompact, 250 intermediate-size cars and 50 SUV’s in the city location and 450 subcompact, 200 intermediate-size cars and 140 SUV’s in suburbs for the month of January.

The Rizza Ford sold 350 subcompact, 100 intermediate-size cars and 30 SUV’s in the city location and 350 subcompact, 300 intermediate-size cars and 100 SUV’s in suburbs for the month of February.

It is provided that the profit on each subcompact car is $\text{\$100}$ , $\text{\$150}$ per intermediate and $\text{\$200}$ per SUV.

The matrix representation of profit on each type of car is $\left[\begin{array}{c}100\\ 150\\ 200\end{array}\right]\begin{array}{c}\text{subcompact}\\ \text{intermediate}\\ \text{SUV}\end{array}$ .

The total sale done in January and February is $\begin{array}{l}\begin{array}{cccccc}& & \text{subcompact}& \text{intermediate}& & \text{SUV}\end{array}\\ \begin{array}{c}\text{city}\\ \text{suburbs}\end{array}\left[\begin{array}{cccccccccc}750& & & & 350& & & & 80& \\ 800& & & & 500& & & & 240& \end{array}\right]\\ \end{array}$ .

Total number of each car sold are,

  $\begin{array}{l}\begin{array}{cccccc}& & \text{subcompact}& \text{intermediate}& & \text{SUV}\end{array}\\ \left[\begin{array}{ccccccccc}& & 1550& & & 850& & & \begin{array}{cc}320& \end{array}\end{array}\right]\end{array}$

Therefore, profit at each location is,

  $\begin{array}{c}\left[\begin{array}{ccc}1550& 850& 320\end{array}\right]\left[\begin{array}{c}100\\ 150\\ 200\end{array}\right]=1550\left(100\right)+850\left(150\right)+320\left(200\right)\\ =346500\end{array}$

Thus, the profit of the company at each location is $\$346500$ .