### Video Transcript

A sheet of metal is cut into a square by a machine that can measure lengths to a resolution of one millimeter. The sides of the sheet are made to be 10 centimeters long. What is the difference between the maximum area that the sheet could have and the minimum area it could have? 0.5 centimeters squared, one centimeter squared, two centimeters squared, 2.5 centimeters squared.

The metal sheet we are interested in is cut by the machine to have side lengths of 10 centimeters. However, we are also told that the resolution of the machine is only one millimeter. What that means is that each of these 10-centimeter lengths is not necessarily exactly 10 centimeters. Recall that the uncertainty in a measurement is half the resolution of the measurement device. Since the resolution is one millimeter, the uncertainty in each of these measurements is half a millimeter. What this means is that each of the 10-centimeter lengths could be as short as 10 centimeters minus 0.5 millimeters or as long as 10 centimeters plus 0.5 millimeters. We write this as 10 centimeters plus or minus 0.5 millimeters.

If we now recall that one millimeter is 0.1 centimeters, we can see that 0.5 millimeters is 0.05 centimeters. And we can concisely write 10 centimeters plus or minus 0.5 millimeters as 10 plus or minus 0.05 centimeters. The reason we have this uncertainty is because a measuring device with a resolution of one millimeter will measure any exact length between 9.95 centimeters and 10.05 centimeters as 10 centimeters. So, when the measuring device returns a value of 10 centimeters, we don’t know for sure that that is the exact length. But we do know for sure that the exact length is no more than half a millimeter different.

Anyway, we’re looking for the difference between the maximum area and the minimum area of the sheet. The maximum area will occur when the length of the sides are the longest, and the minimum area will occur when the length of the sides are the smallest. The maximum side length consistent with a measurement of 10 centimeters and a measurement resolution of one millimeter is 10 plus 0.05 centimeters. So the side lengths of the sheet with the maximum area are 10.05 centimeters. Since the area of a square is the square of the length of its sides, the maximum area the sheet can have is 10.05 centimeters squared, which is 101.0025 square centimeters.

Following the same progression, the minimum side length consistent with a measurement of 10 centimeters and a measurement resolution of one millimeter is 10 minus 0.05 centimeters. This is a side length of 9.95 centimeters, and 9.95 centimeters squared is 99.0025 square centimeters. We have found then that the maximum possible value for the area of the sheet is 101.0025 centimeters squared. And the minimum possible value for the area of the sheet is 99.0025 centimeters squared. And the difference between these two values is two centimeters squared. And two centimeters squared is the answer that we are looking for.

It’s worth taking a brief moment to compare the percent uncertainty in the length of the sides to the percent uncertainty in the area of the sheet. As we can see from the maximum and minimum values for the side lengths, the percent uncertainty is one-half of one millimeter into 10 centimeters, which is one-half of one percent. If the side length of the sheet is exactly 10 centimeters, then the area of the sheet is exactly 100 centimeters squared. So, from the maximum and minimum possible values for the area of the sheet, we can see that the uncertainty is about one centimeter squared. So the percent uncertainty is one centimeter squared in 100 centimeters squared, which is one whole percent.

One whole percent, the uncertainty in the area of the sheet, is larger than one-half percent, the uncertainty in the side lengths of the sheet. Note also that to calculate the area of the sheet, we had to multiply two side lengths. So we had to multiply two quantities with uncertainty. This illustrates an important general principle. The percent uncertainty in the product of two quantities that themselves have uncertainty is larger than the percent uncertainty of the individual factors.