Video Transcript
A 20-centimeter-long measuring stick has 50 evenly spaced lines marked along its length. What is the resolution of the measuring stick in millimeters?
To answer this question, we will first need to understand the resolution of a measuring device.
In a very general sense, the resolution of a measuring instrument is the minimum difference between two different objects so that we can always detect that those two objects have different values for the property we are measuring. For example, on a measuring stick, like the one in our question, the resolution is the distance between two adjacent marks.
To see this visually, we’ve drawn two objects. The exact length of the magenta spotted object is just a little bit longer than two and a half marks. And the exact length of the striped orange object is just a little bit shorter than three and a half marks.
Since both of these lengths are closer to three full marks than to any other whole number of marks, we would measure both of the lengths of these objects as three marks. However, as we can see, the difference between the true lengths of these objects is actually almost the full resolution of the measuring stick. So we see that there are situations where objects can differ by up to a full mark and still be measured to have the same length.
However, if the difference in the lengths of the two objects is greater than one mark, that is, greater than the resolution, we would always measure their lengths to be different. Here, we’ve made the striped object just a little bit longer. So the difference between the true lengths of the objects is greater than the resolution. Because the difference in length is greater than one mark, the exact length of the yellow striped object is now a little bit larger than three and a half marks, which as we know we would measure as four marks.
So we now measure a full mark of difference between the lengths of the two objects. In other words, we measure them to have different lengths. And this is only possible because the difference between their lengths is greater than the resolution of the measuring stick.
Okay, so to find the resolution we’re looking for, we need to find the distance between two adjacent marks. And we know how long the stick is overall and also how many marks there are and that they are evenly spaced. This reduces the problem to simple division. If there are 50 evenly spaced marks along the stick and the total length of the stick is 20 centimeters, then the distance between two adjacent evenly spaced lines is the length of the stick divided by the total number of lines. So the distance between two adjacent marks, that is, the resolution of the measuring stick, is 20 centimeters divided by 50. So the resolution of the stick is 0.4 centimeters.
This is almost the answer we want. But we are told to present this answer in millimeters. We recall that one millimeter is 0.1 centimeters. So 0.4 centimeters would just be four millimeters. So the answer we’re looking for, the resolution of our measuring stick that is 20 centimeters long with 50 evenly spaced lines, is four millimeters.
It’s interesting to note that four millimeters is two percent of 20 centimeters. So we can also say that the resolution of the measuring stick is two percent of the overall length. This is actually entirely logical. We divided the stick into 50 equally sized pieces, and the resolution is the size of one of those pieces. So the resolution is one fiftieth of the overall length of the stick, and one fiftieth is two parts in 100, which is two percent.
This is a useful way of thinking because it allows us to conclude that no matter how long our measuring stick is, if it is marked with 50 evenly spaced lines, the resolution will be two percent of the overall length of the stick.
