Video Transcript
A measuring stick and a measuring tape both have marks separated by one millimeter. The stick has 300 of these marks, and the tape has 5000 of these marks. What is the advantage of using the tape for measuring lengths? The tape has greater resolution. The tape can measure lengths greater than 300 millimeters. All of the answers are correct.
In this question, we are asked about two different tools that measure length. The first is a measuring stick, and the second is a measuring tape. We’re also told that both the tape and the stick have a series of marks. And the distance between each mark and its nearest neighbors is one millimeter. The difference between the stick and the tape is that the stick has 300 marks in total, while the tape has 5000 marks in total. Note that we have exaggerated the scale of the mark somewhat to make it easier to visualize. If we really were trying to fit 300 marks on the stick or 5000 marks on the tape, we would either need to make this stick and the tape much longer or the marks much closer together.
Anyway, we are specifically trying to determine what advantage the tape has over the stick when it comes to measurement resolution and the maximum length that each tool can measure. Now recall that to measure the length of an object, we align one end of the object with one of the marks on our measuring tool and then count the number of marks until we reach the other end of the object. Since we count the number of marks until we reach the end of the object, in order to properly find the length of our object, the portion of our measuring tool with marks on it must be at least as long as the object itself.
Since the measuring stick has 300 marks that are each one millimeter apart, the total length of the measuring stick with marks on it is 300 millimeters. The tape also has marks that are separated by a distance of one millimeter, but the tape has 5000 marks. So the total length of the tape that is useful for measuring objects is 5000 millimeters. Comparing these two values, we see that the tape is longer than the stick. So the tape can measure longer lengths than the stick can. This is a qualitative comparison.
However, we are asked to make a quantitative comparison specifically whether or not each of these tools can measure lengths that are greater than 300 millimeters. The useful part of the tape is 5000 millimeters long. 5000 millimeters is much greater than 300 millimeters. So the tape can certainly measure lengths that are greater than 300 millimeters. However, the useful part of the stick is only 300 millimeters long, which means that the maximum length that the stick can measure is 300 millimeters. So option (B) is indeed an advantage of the measuring tape over the measuring stick.
The useful length of the measuring tape is longer than 300 millimeters. So it can measure lengths greater than 300 millimeters. However, the useful length of the stick is only 300 millimeters, so it cannot measure lengths greater than 300 millimeters. Notably, however, option (B) would not be an advantage if we were looking at lengths that were less than 300 millimeters because both the tape and the stick are long enough to measure any object that is less than 300 millimeters long.
All that’s left now is to determine whether the tape has greater resolution than the stick. One way to think about the resolution of a measuring tool is that it is the smallest difference that that tool can always detect. To see what we mean by this, note that the object we drew before does not align exactly to the marks on the measuring stick. So when measuring this object, we round to the nearest mark. So we would measure this object as being a full three marks long, even though its true length is between two and three marks long.
In fact, because we are rounding to the nearest whole number of marks, anything between half a mark before and half a mark after the third mark will be measured as three marks. The difference between three and half marks and two and half marks is one full mark. So the maximum range of true lengths that we would still measure as the same length is exactly one mark wide. In other words, something that was originally two and half marks could grow by up to one full mark before we would measure it differently on the measuring stick. Therefore, one full mark is the smallest difference that we will always be able to detect with our measuring stick.
The distance between two marks on the measuring stick is one millimeter, so the resolution of the measuring stick is one millimeter. The distance between two marks on the tape is also one millimeter, so the resolution of the tape is also one millimeter. That is, the stick and the tape have the same resolution. So (A) is not an advantage of the tape over the stick because the tape and the stick have the same resolution.
Of our answer choices then, we see that the only advantage that the tape has over the stick when it comes to measuring lengths is that the tape can measure lengths greater than 300 millimeters.
