Chapter 12.2, Problem 35PPSE

Solution

a.

To calculate: The matrix to determine the score in the track meet if two teams are competing in two-team track meet. Points for individual events are awarded as follows: 5 points for first place, 3 points for second place, and 1 point for third place. Points for relays are awarded as follows: 5 points for first place, and no points for second place.

The total score for west river team is 97 points and River’s edge team is 99 points.

Given information:

Two teams are competing in two-team track meet. Points for individual events are awarded as follows: 5 points for first place, 3 points for second place, and 1 point for third place. Points for relays are awarded as follows: 5 points for first place, and no points for second place.

Formula Used:

Use the matrix multiplication.

Calculation :

It can be seen that points for west river are awarded as follows: 5 points for first place, 3 points for second place, and 1 point for third place.

Use matrix multiplication to calculate the score.

  $\begin{array}{c}\left[\begin{array}{ccc}8& 5& 2\end{array}\right]\left[\begin{array}{c}5\\ 3\\ 1\end{array}\right]=8\left(5\right)+5\left(3\right)+2\left(1\right)\\ =40+15+2\\ =57\end{array}$

It can be seen that points for team relay are awarded as follows: 5 points for first place, and no points for second place.

Use matrix multiplication to calculate the score.

  $\begin{array}{c}\left[\begin{array}{cc}8& 5\end{array}\right]\left[\begin{array}{c}5\\ 0\end{array}\right]=8\left(5\right)+5\left(0\right)\\ =40+0\\ =40\end{array}$

Total score for west river team is $57+40=97$ points.

The points for River’s are awarded as follows:

Individual events: 5 points for first place, 3 points for second place, and 1 point for third place.

Use matrix multiplication to calculate the score.

  $\begin{array}{c}\left[\begin{array}{ccc}6& 9& 12\end{array}\right]\left[\begin{array}{c}5\\ 3\\ 1\end{array}\right]=6\left(5\right)+9\left(3\right)+12\left(1\right)\\ =30+27+12\\ =69\end{array}$

Team Relay: 5 points for first place, and no points for second place.

  $\begin{array}{c}\left[9\right]\left[\begin{array}{c}5\\ 0\end{array}\right]=6\left(5\right)+9\left(0\right)\\ =30+0\\ =30\end{array}$

Total score for River’s edge team is $69+30=99$ points.

b.

To calculate: The matrix to determine the score in the track meet if two teams are competing in two-team track meet. Points for individual events are awarded as follows: 5 points for first place, 2 points for second place, and 1 point for third place. Points for relays are awarded as follows: 5 points for first place, and no points for second place.

The total score for west river team is 92 points and River’s edge team is 90 points.

Given information:

Two teams are competing in two-team track meet. Points for individual events are awarded as follows: 5 points for first place, 2 points for second place, and 1 point for third place. Points for relays are awarded as follows: 5 points for first place, and no points for second place.

Formula Used:

Use the matrix multiplication.

Calculation :

It can be seen that points for west river are awarded as follows: 5 points for first place, 2 points for second place, and 1 point for third place.

Use matrix multiplication to calculate the score.

  $\begin{array}{c}\left[\begin{array}{ccc}8& 5& 2\end{array}\right]\left[\begin{array}{c}5\\ 2\\ 1\end{array}\right]=8\left(5\right)+5\left(2\right)+2\left(1\right)\\ =40+10+2\\ =52\end{array}$

It can be seen that points for team relay are awarded as follows: 5 points for first place, and no points for second place.

Use matrix multiplication to calculate the score.

  $\begin{array}{c}\left[\begin{array}{cc}8& 5\end{array}\right]\left[\begin{array}{c}5\\ 0\end{array}\right]=8\left(5\right)+5\left(0\right)\\ =40+0\\ =40\end{array}$

Total score for west river team is $52+40=90$ points.

The points for River’s are awarded as follows:

Individual events: 5 points for first place, 2 points for second place, and 1 point for third place.

Use matrix multiplication to calculate the score.

  $\begin{array}{c}\left[\begin{array}{ccc}6& 9& 12\end{array}\right]\left[\begin{array}{c}5\\ 2\\ 1\end{array}\right]=6\left(5\right)+9\left(2\right)+12\left(1\right)\\ =30+18+12\\ =60\end{array}$

Team Relay: 5 points for first place, and no points for second place.

  $\begin{array}{c}\left[9\right]\left[\begin{array}{c}5\\ 0\end{array}\right]=6\left(5\right)+9\left(0\right)\\ =30+0\\ =30\end{array}$

Total score for River’s edge team is $60+30=90$ points.