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**Topic**

A football field is rectangular in shape with the length and width of the field being 105 m and 68 m respectively. The longest distance between two positions on the field is exactly equal to the diagonal length of the field. Find an approximation (in meters) of the diagonal length of the yard and find the accuracy and relative error of that approximation.

**Solution method – See details**

Let \(x\) be the diagonal length of the football field. Calculate \(x\) and find the precision and relative error of \(x\)

**Detailed explanation**

Let \(x\) be the diagonal length of the football field. Applying the Pythagorean theorem, we have:

\(x = \sqrt {{{105}^2} + {{68}^2}} = \sqrt {15,649} = 125,09596…\)

Taking an approximate value of \(x\) of 125.1, we have: \(125.09 < x < 125.1\)

\( \Rightarrow \left| {x – 125.1} \right| < \left| {125.09 – 125.1} \right| = 0.01\)

So the length of the football field can be taken as 125.1 with precision \(d = 0.01\)

The relative error of 125.1 is \({\delta _{125.1}} = \frac{{{\Delta _{125.1}}}}{{\left| {125.1} \right |}} < \frac{{0.01}}{{125.1}} \approx 0.08\% \)

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