The graph shows the relationship between distance and time in a 160-metre race between a hare and a tortoise. Determine the winner and then find the average speeds of the hare and the tortoise.
Okay, so to start solving this problem, there’s obviously two parts to it. The first part is actually what we’ve got to do is determine the winner. Okay, right, so now, we’re gonna have a look to see who wins the race.
In order to see who wins the race, what does it actually mean? What’s winning the race mean? Well, winning the race actually means finishing the race in the quickest time. So let’s have a look from the graph how long it took the tortoise and how long it took the hare to finish the race.
Remarked here on the graph that 160 metres, which is the end of the race, it’s taking the tortoise 70 minutes to complete the race. Now, take a look at the hare and see how long it took the hare to finish the race.
And as you can see here marked on the graph, after 160 metres. So at the end of the race it’s taking the hare 90 minutes. So therefore, the tortoise finished the race 20 minutes faster than the hare. So the tortoise is the winner.
Okay, so we’ve now solved the first part of the problem. And we’ve determined who’s won the race. We know the tortoise is the winner. So now, we’re gonna have a look at finding the average speeds of both the hare and the tortoise. I’m gonna leave the marks on the graph cause they’re gonna be very useful.
In order to find the average speed of the hare and the tortoise, we’re gonna use something to help us that actually comes in from physics as well as this subject. This is the speed-distance-time triangle, and what it tells us is actually a good way of remembering the formula for speed, distance, and time.
So for instance, we would have speed is equal to distance over time. And this is because when we look at the triangle, we could see that the 𝑑 is above the 𝑡. So that means it’s gonna be distance divided by time.
Similarly, we can see that time is equal to distance over speed or distance divided by speed. And this again is because the time would be on its own. And then you’ve got the distance over this 𝑠. So it’s divided by the 𝑠. And it gives us our second equation.
And the final equation we can get from our triangle is that distance is equal to speed multiplied by time. And this is because the distance is above the other two. And the actual speed and time are next to each other. So that means multiplied by each other. So there we go. We’ve got the three little equations that could help.
In this one, we’re gonna be using the top equation because we’re looking at speed. So we’re now gonna use this equation to help us find the average speed. And we would start with the hare. We’ve slightly adapted the equation as you can see cause it’s average speed. That means it’s going to be the total distance divided by the total time because there are different sections on a graph. But it’s the total distance divided by the total time.
So let’s work out the average speed of the hare. Okay, so we’re gonna look at the average speed. It’s gonna be equal to total distance or the total distance is a 160 metres because that’s the length of the race. And the total time — well, we actually looked at that earlier because we’ve marked it already on the graph — is 90 minutes.
Great! We’re now at a stage where we can actually simplify this to find our average speed. It can serve us for we divide the numerator and denominator by 10, which gives us 16 over nine. But to leave it in a better format, we’ll convert it to a mixed number which’s given as one and seven-ninths.
But the key point to remember here: do not forget the units, which are metres per minute. Great! So we’ve now found the average speed of the hare, we can move on to the average speed of the tortoise. Again, we can use the same formula as before. So we have the average speed equals the total distance over total time. Again, the total distance is 160 metres cause it’s the length of the race. And the total time when we look back to the graph where we already marked it, we’ll see that it’s 70 minutes.
We’ll then simplify again by dividing numerator and denominator by 10, which gives 16 over seven. And as we did before, just to give it a better form, we’ll convert that to a mixed number, which gives us two and two-sevenths. And that’s again because there are two-sevenths going into 16 with a remainder of two, so two and two-sevenths.
And don’t forget the units; that’s metres per minute. Fantastic! As we’ve said, we’ve calculated the average speed of the hare, the average speed of the tortoise and decided who’s the winner.
But we can quickly check to make sure that it all seems to make sense. And we can do that because we know the tortoise is the winner cause it finished the race in the quickest time. So when we look at the average speeds, the tortoise has the highest average speed, which makes sense cause it’s the winner. If we didn’t have that, we would know that there’s something wrong with our calculation.