The graph represents the distance traveled by a speed boat. Use the graph to determine, in miles per hour, the rate of change in distance.
And then we have a graph which represents the distance traveled, miles, at time ℎ, hours. Let’s first begin by recalling what we mean by rate of change when it comes to graphs. The rate of change is found by considering the slope or the gradient of the graph. But of course, here we’re dealing with a distance–time graph. And when we find the slope on a distance–time graph, we’re actually finding the speed at that point. And so, the rate of change of distance of our graph will represent the average speed of the speed boat.
And so, we actually have two ways to answer this question. One is to find the slope or the gradient of the line, and the other is to use the formula for average speed. Of course, those will give us the same result, so let’s have a look at both methods.
We’ll begin by using the formula average speed is distance divided by time. We’re dealing with a straight line. And so the average speed is going to be consistent throughout the journey. This means we can pick any point on this line. A really sensible one is the point with coordinates one, 80. The distance at this point is 80 miles and the time taken is one hour. So, the speed here is 80 divided by one, which is simply 80. Now, of course, we’re working with miles and hours, so the speed must be 80 miles per hour.
That, of course, is also equal to the rate of change of distance. But let’s briefly look at how we could’ve calculated that by considering the slope of our graph. The formula to find the slope is rise over run or change in 𝑦 divided by change in 𝑥. And so, we pick two coordinates on the graph. Zero, zero is a sensible one. And we could have chosen one, 80, but for argument’s sake, let’s choose a different point. Let’s choose the point with coordinates 0.5, 40.
Change in 𝑦 is then the vertical distance; it’s 40. And change in 𝑥 is the horizontal length of the line, so it’s 0.5. So, the slope, change in 𝑦 divided by change in 𝑥, is 40 divided by 0.5. Now, it’s really useful to remember here that dividing by 0.5 is the same as timesing by two. So, this calculation is 40 times two, which is 80. And so, we can say that the rate of change of distance of the speed boat, which is also speed or slope, is 80 or 80 miles per hour.