Chapter 3.6, Problem 35PPS

a.

To determine

To find: A matrix to determine which sales associate earned the most money.

Answer to Problem 35PPS

Matrix to determine which sales associate earned most is

  $E=\left[\begin{array}{c}51,500\\ 53,000\\ 37,000\\ 46,000\end{array}\right]$

Explanation of Solution

Given equation:

Each sales associate gets a bonus of $\$1000$ for every new car they have sold.

And $\$500$ for every used car they have sold.

Table represents Cars sold by each associate.

From the given table

We can write a matrix for number of cars sold by each associate.

Rows represents number of different types of Sales associates.

First column represents the number of new cars.

Second column represents the number of used cars.

Therefore,

Number of columns will be 2

Number of rows will be 4

Therefore,

Matrix for number of cars sold by each associate is

  $N=\left[\begin{array}{cc}27& 49\\ 35& 36\\ 9& 56\\ 15& 62\end{array}\right]$

From given information

Each sales associate gets a bonus of $\$1000$ for every new car they have sold.

And $\$500$ for every used car they have sold.

There will be two rows, first row represents bonus for new car and second row represents bonus for selling used car.

Bonus matrix is

  $B=\left[\begin{array}{c}1000\\ 500\end{array}\right]$

Therefore,

Matrix for earning of sales associate is

  $\begin{array}{l}E=\left[\begin{array}{cc}27& 49\\ 35& 36\\ 9& 56\\ 15& 62\end{array}\right]\cdot \left[\begin{array}{c}1000\\ 500\end{array}\right]\\ \Rightarrow E=\left[\begin{array}{c}\left(27\times 1000\right)+\left(49\times 500\right)\\ \left(35\times 1000\right)+\left(36\times 500\right)\\ \left(9\times 1000\right)+\left(56\times 500\right)\\ \left(15\times 1000\right)+\left(62\times 500\right)\end{array}\right]\\ \Rightarrow E=\left[\begin{array}{c}27000+24500\\ 35000+18000\\ 9000+28000\\ 15000+31000\end{array}\right]\\ \Rightarrow E=\left[\begin{array}{c}51,500\\ 53,000\\ 37,000\\ 46,000\end{array}\right]\end{array}$

Therefore,

Matrix to determine which sales associate earned most is

  $E=\left[\begin{array}{c}51,500\\ 53,000\\ 37,000\\ 46,000\end{array}\right]$

b.

To determine

To find: Total amount of money the car lot spent on bonuses for the sales associates this year.

Answer to Problem 35PPS

Total amount of money the car lot spent $=\$187,500$

Explanation of Solution

Given equation:

Each sales associate gets a bonus of $\$1000$ for every new car they have sold.

And $\$500$ for every used car they have sold.

Table represents Cars sold by each associate.

Matrix to determine which sales associate earned most is

  $E=\left[\begin{array}{c}51,500\\ 53,000\\ 37,000\\ 46,000\end{array}\right]$

Here each row represents the income earned by sales associate earned of four different types.

Therefore,

Total amount of money the car lot spent $=51,500+53,000+37,000+46,000$

  $\Rightarrow$ Total amount of money the car lot spent $=\$187,500$