Chapter 12.1, Problem 21P

Race times, in seconds, for four members of the track team are shown below. For each distance run, which runner showed the greatest improvement, in seconds, from the preliminary race to the final race? Find the answers using matrices.

To determine

For each distance run, find the runner that showed the greatest improvement , in seconds, from the preliminary race to the final race .

Answer to Problem 21P

Moore had greatest improvement in 200m race while Chandra had the greatest improvement in the 400m race.

Explanation of Solution

Given:

  

Formula Used:

  $\left[\begin{array}{l}\text{a b }\\ \text{d e}\\ \text{g h}\end{array}\right]-\left[\begin{array}{l}\text{j k}\\ \text{m n}\\ \text{p q}\end{array}\right]=\left[\begin{array}{l}\text{a-j b-k}\\ \text{d-m e-n}\\ \text{g-p h-q}\end{array}\right]$

  $\left[\begin{array}{l}7.52\\ 7.02\\ 4.24\\ 7.94\end{array}\right]$

Calculation:

Arrange the Preliminary race and Final race in matrices:

Preliminary race

  $\left[\begin{array}{l}\text{ 200m 400m}\\ Haddock\text{ 24}\text{.69 57}\text{.02}\\ Romano\text{ 25}\text{.53 61}\text{.16}\\ Chandra\text{ 25}\text{.56 59}\text{.67}\\ \text{Moore 24}\text{.81 58}\text{.11}\end{array}\right]$

Final race

  $\left[\begin{array}{l}\text{ 200m 400m}\\ Haddock\text{ 24}\text{.61 58}\text{.08}\\ Romano\text{ 25}\text{.52 60}\text{.38}\\ Chandra\text{ 25}\text{.63 57}\text{.72}\\ \text{Moore 24}\text{.72 58}\text{.52}\end{array}\right]$

Subtrace the Final race timings from the preliminary race timings to find the improvements of the runners.

  $\left[\begin{array}{l}\text{24}\text{.69 57}\text{.02}\\ \text{25}\text{.53 61}\text{.16}\\ \text{25}\text{.56 59}\text{.67}\\ \text{24}\text{.81 58}\text{.11}\end{array}\right]-\left[\begin{array}{l}\text{24}\text{.61 58}\text{.08}\\ \text{25}\text{.52 60}\text{.38}\\ \text{25}\text{.63 57}\text{.72}\\ \text{24}\text{.72 58}\text{.52}\end{array}\right]=\left[\begin{array}{l}\text{0}\text{.08 -1}\text{.06}\\ \text{0}\text{.01 0}\text{.78}\\ \text{-0}\text{.07 1}\text{.95}\\ \text{0}\text{.09 -0}\text{.41}\end{array}\right]$

So, larger the value of the difference, greater the improvement in the runners .In first column , the greatest value is 0.09 and in the second column , the greatest value is 1.95.

So, Moore had greatest improvement in 200m race while Chandra had the greatest improvement in the 400m race.