The graph shows a plot of change in distance over time. What is the unit of the gradient of the line? What is the unit of the area under the line?
Okay, so in this question, we’ve got this graph, which is showing us the change in distance on the vertical axis and the change in time on the horizontal axis. What we’re asked to do is to find the unit of the gradient of the line and the unit of the area under the line.
So let’s start by finding out the unit of the gradient of the line. Well, first of all, how do we find the gradient of a straight line? Well, what we can do is to pick two points on the line — say this point here and this point here — and we calculate their rise over run. And we calculate the rise over run.
The rise is how much change there is in the vertical axis, so what these values are here, and the run is how much change there is in the horizontal axis, so these values here. Now, when we’re looking at the change in the vertical coordinates, so we’re looking at the rise, this rise is going to be in metres because the unit on the vertical axis is metres.
And so when we find out the rise, what we’re basically doing is finding the difference in the distance between the two points. Hence, the unit of the rise is going to be in metres. Similarly, when we’re calculating this run here, what we’re finding out is the difference in time because these values here are going to be in seconds. So the unit of the run is also going to be seconds.
And hence, for the gradient, we’ve got rise over run. And we saw that rise has a unit of metres and the run has a unit of seconds. Therefore, the gradient of the line will have the unit metres per second or metres divided by seconds. Because when we calculate the gradient of a straight line, we take two points on the line and then calculate the difference in the vertical coordinates of the two points divided by the difference in the horizontal coordinates of the two points. So the unit for the gradient is metres per second.
Now, let’s look at the unit for the area under the line. Now, if we want to calculate the area under the line, then normally we’ll do this for two limiting horizontal axis coordinates. Basically, what that means is that we’d find the area under the line for two different times in this case because time is on the horizontal axis.
So let’s say we’re trying to find the area under the line for this time which is zero seconds and this time which is well whatever it may be. In that case, we’re trying to find out this area here. Now, the choices of time are arbitrary. We could have, for example, chosen somewhere between this time and this time for example. So we would have wanted to work out this area here, but it makes it easier to work with a triangle.
And anyway, we’re only trying to find out the units. And regardless of which area we choose, the units would always be the same. So let’s work with the simplest shape that we have. Let’s work with the triangle. So because the units of the area are the same regardless of which two points we choose for the time, let’s just work with the simplest shape that we have; let’s work with a triangle. And let’s just work out the orange-shaded area.
So because we’ve got a triangle, we need to work out the area of a triangle. We can recall that the area of a triangle is a half multiplied by the base length multiplied by the height. In other words, half multiplied by this length here, the base length, multiplied by the height.
Now, in this case, the half is kind of irrelevant because it doesn’t contribute to the units. However, we know that for the base length, it has a unit of seconds and for the height it has a unit of metres. Because the base length is measuring from zero seconds all the way to whatever time this may be in seconds and the height is measuring from zero metres to whatever this distance maybe in metres, therefore, when we multiply them together, we find that the area of a triangle has the units of well half doesn’t have a unit multiplied by units of base which is seconds multiplied by the units of height which is metres.
In other words, the unit of the area under the line is metres multiplied by seconds and that is our final answer.
So the unit of the gradient of the line is metres per second and the unit of the area under the line is metres multiplied by seconds.