91Ó°ÊÓ

When we talk about measurement error, we refer to the uncertainty or deviation in the measured value compared to the actual or true value. In our scenario, we are concerned with the measurement error in estimating the side length of a square floor tile. While the side is estimated to be 1 foot, there's a possible error of \(\pm \frac{1}{16}\) inch. To simplify calculations, it's vital to convert this inch-based error into feet, since our original measurement is in feet. We know there are 12 inches in a foot. Therefore, \(\frac{1}{16}\) inch translates into \(\frac{1}{16 \times 12} = \frac{1}{192}\) feet, giving us a clearer understanding of the extent of the potential error in terms of our current unit of measurement.

The area of a square tile is computed using the formula \(A = s^2\), where \(s\) represents the length of one side of the square. For our tile, the side length \(s\) is 1 foot—a straightforward case, as the area calculation simplifies to \(1^2 = 1\) square foot. However, when considering measurement errors, we need to assess how these uncertainties in length impact the computed area.To do this, we employ the concept of differentials. A differential helps us estimate how a small change in one variable, the side length \(s\), affects another, the area \(A\). The change in area, denoted as \(dA\), can be estimated using the formula \(dA = 2s \, ds\), whereby \(ds\) is the change in the side length. This mathematical approach gives a tangible sense of how a slight fluctuation in side measurement affects the total area.

Percentage error is a useful metric to quantify the scale of error relative to the actual value. In practical terms, it indicates the proportion of the error in comparison to the true measurement, expressed as a percentage. For our square tile problem, we previously calculated the differential change in area as \(dA = \frac{1}{96}\) square foot, based on the slight error in side length. Since the exact area without errors is 1 square foot, the percentage error is computed as:\[\frac{\text{Error in Area}}{\text{Exact Area}} \times 100\% = \frac{1/96}{1} \times 100\% = \frac{100}{96}\% \approx 1.04\%\]This calculation highlights that, despite the small absolute error, the relative error is about 1.04%, providing a succinct measure of the error's magnitude in context of the total area.