Video Transcript
Scarlett uses a centimeter ruler to
measure the length of a straight line as shown in the diagram. She determines that the length of
the line is 10.4 centimeters. Which of the following statements
explains why this answer is incorrect? A) The maximum resolution of the
ruler is one centimeter. Thus, the length of the line should
be recorded as 10 centimeters. B) Measurements using a ruler
should always be rounded up. Thus, the length of the line should
be recorded as 11 centimeters. C) The ruler is not parallel to the
line. Thus, the line is actually shorter
than 10.4 centimeters. D) The ruler is not parallel to the
line. Thus, the line is actually longer
than 10.4 centimeters.
Okay, so what Scarlett’s done here
is try to measure the length of this line here. The line is labeled 𝑥. And she’s wrote off using the ruler
that the length of the line is 10.4 centimeters. Now this is an incorrect
measurement of the length of the line. And we need to work out why that
is.
So looking at option A as a
possible answer, this one says that the maximum resolution of the ruler is one
centimeter. Thus, we should record the length
of the line as 10 centimeters rather than 10.4. Well, we’ve been told in the
initial part of the question that Scarlett uses a centimeter ruler. This means that the large markings
on the ruler are every one centimeter. And hence, the smaller markings
must be every tenth of a centimeter or every millimeter. This means that the maximum
resolution of the ruler is not in fact one centimeter. It’s one millimeter. And hence, option A can be ruled
out. Haha, ruled out.
Moving on to option B then, this
one says that measurements using a ruler should always be rounded up. Now this is not a very good idea at
all because always rounding up a measurement will introduce a systematic error in
our whole set of measurements.
Let’s say we’re trying to measure
the lengths of different lines all over the place. If we always round up their
measurements, then unless one of these lines falls exactly on a perfect centimeter
marking, we’re going to approximate them to be longer than they actually are. Because we’ll say that they’re
rounded up. Now this is not a good idea in
terms of an entire sample. Because the way that rounding
normally works is that, on average, 50 percent of the time we will round up. So we will assume that the length
of a line is longer than it actually is. And 50 percent of the time we will
round down, just as we do with normal rounding. And so any rounding up done in some
lines is canceled by the rounding down done in other lines.
However, if we always round up when
we make measurements with a ruler, then we’re always overestimating the length of a
line, unless, like we said, it falls perfectly on a centimeter marking. So, for example, if a line is
measured to be exactly four centimeters, then we wouldn’t need to round up. But overall, we’re still
overestimating the lengths of lines in our sample. So this is not a good thing. And hence, option B is out of the
question as well.
Let’s then look at option C. This one says that the ruler is not
parallel to the line. Thus, the line is actually shorter
than 10.4 centimeters. And actually, if we look at option
D, it starts off the same way as option C. It says that the ruler is not
parallel to the line. Where it differs is in the
consequence.
Option D says that the line is
actually longer than 10.4 centimeters. So because these are the only two
options left, we can safely say that the reason that Scarlett’s measurement is
incorrect is because the ruler and the line are not parallel. And that actually is the reason
why. Because if we want to measure the
length of the line, then we need to be measuring from here to here. And hence, the ruler and the line
need to be parallel with each other.
So to work out the consequence of
this, let’s imagine that we rotate this line clockwise until it’s parallel with the
ruler. Or if we want to, we can imagine
rotating the ruler counterclockwise. It doesn’t matter which as long as
we rotate one or both until they’re parallel with each other.
So now this is what the line would
look like if we rotated it until it was parallel with the ruler. In this case, we can see that the
left-hand end of the line is the point about which we rotated it. And the reason we’ve done this is
because the left-hand end of the line is aligned perfectly with the zero marking on
the ruler. This is a good thing. This is something that Scarlett has
done well. Because in order to measure the
length of the line once the line is parallel with the ruler, one end needs to be
aligned perfectly with the zero marking on the ruler. And the other end is used to take
the measurement.
So we rotate about the left-hand
end and then take the measurement on this end. And we can see that the measurement
is actually roughly 11 centimeters. The exact measurement doesn’t
matter. But the point is that the actual
measurement of the line when the line is parallel with the ruler is larger than 10.4
centimeters. And hence, we’ve worked out that
the line is actually longer than 10.4 centimeters. That’s what option D says. Therefore, option C is
incorrect.
And so we have the final answer to
our question. The reason that Scarlett’s
measurement of 10.4 centimeters is incorrect is because the ruler is not parallel to
the line. Therefore, the line is actually
longer than 10.4 centimeters.
