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Question

The length and width of a rectangular room are measured to be $3.955 \pm 0.005 \textrm{ m}$ and $3.050 \pm 0.005 \textrm{ m}$. Calculate the area of the room and its uncertainty in square meters.

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Final Answer

$12.06 \pm 0.04 \textrm{ m}^2$

Solution Video

OpenStax College Physics Solution, Chapter 1, Problem 27 (Problems & Exercises) (2:43)

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Video Transcript

This is College Physics Answers with Shaun Dychko. We're given the length and width of a room and our job is to find its area and then express that area with uncertainty as well. So, the length is 3.955 plus or minus 0.005 meters, and the width is 3.05 plus or minus 0.005 meters. So, the area is length times width, and we do this calculation without thinking about the uncertainties because we'll do that in the next step. So, we have 3.955 meters times 3.05 meters giving us 12.06275 square meters. And we're not sure what precision to round that to yet, but we’ll calculate the uncertainty and then the uncertainty will tell us what the precision should be in this calculation. The percent uncertainty in a... answer of multiplying or dividing can be found by adding the percent uncertainties in the factors. So, because we're multiplying length by width here, we can find the percent certainty in area by adding the percent uncertainty in the length to the percent uncertainty in the width. And we have <i>L</i>, <i>W</i> and <i>A</i> there to label each of these percent uncertainties. So, the percent uncertainty in the length is the absolute uncertainty in the length of .005 meters divided by the length of 3.955 meters and then times that by 100% And then, we have percent uncertainty in the width is also .005 meters, but then it's divided by 3.05 meters and that gives .29%. And then, the absolute uncertainty in the area will be the percent certainty multiplied by the area, and that is .29 over a 100 because percent can always be written as over 100 because if you know French, the word cent is also the word 'cent' in French, which means the number 100. And so, percent can be thought of as per 'cent', which means per 100. So, the numbers over 100. So, we have .29 divided by a 100 multiplied by 12.06275 and that gives .035 square meters which we can round to only one significant figure since uncertainties are not precise. And so, we have .04 square meters and this means our area calculation should be rounded to the hundreds place because that is the precision of our uncertainty. And so, our final answer for area is 12.06 plus or minus .04 square meters.