Video Transcript
A curved racetrack has a length of 440 meters. How many times does an athlete running at 8.8 meters per second cross the same point on the racetrack if they run for 250 seconds?
We are told that the racetrack has a length of 440 meters; i.e., the distance that the athlete has to run in order to complete one lap of the track is 440 meters. We are told that the speed of the athlete is 8.8 meters per second and that they run for 250 seconds. The question asks us how many times they cross the same point on the racetrack. Now, there are actually two parts to this question. First, we need to calculate the total distance that the athlete runs. Then, we need to convert this total distance into a number of times that they pass the same point on the racetrack.
In order to work out the distance that the athlete runs, we should recall that speed is equal to distance divided by time. We’ll call the speed 𝑠, the distance 𝑑, and the time 𝑡. Then we can write that 𝑠 is equal to 𝑑 divided by 𝑡. The question gives us the speed of the athlete, that’s 8.8 meters per second, and the time for which they are running, 250 seconds. So we have values for both 𝑠 and 𝑡.
In order to calculate the distance, we need to rearrange our equation for the speed in order to make the distance 𝑑 the subject. To do this, we multiply both sides of the equation by 𝑡. Then the 𝑡’s in the numerator and denominator on the right-hand side of the equation cancel out. Then we can write that distance 𝑑 is equal to speed 𝑠 multiplied by time 𝑡. Notice that we have swapped the left- and the right-hand sides of the equation.
Now we need to substitute in our values for 𝑠 and 𝑡. Doing this, we have that the distance 𝑑 is equal to 8.8 meters per second multiplied by 250 seconds. This gives us a distance of 2200 meters. So that’s step one complete. We can check that one off. We now just have step two left to go. We are asked how many times the athlete will cross the same point on the track. So we need to know how many laps of the track this distance of 2200 meters is equal to. To make it clear where the number of times the athlete crosses the same point on the track is equivalent to finding the number of laps that the athlete runs, let’s consider a given point on the track.
During one lap of the track, the athlete crosses this point precisely once. During a second lap of the track, the athlete crosses this point once more and so on. For every additional lap, the athlete crosses this point one more time, so the number of laps run is equal to the number of times that the athlete crosses the point. To calculate the number of laps, we observe that the total distance that the athlete runs must be equal to the number of laps that they complete multiplied by the distance covered in one lap. This total distance is 𝑑, and let’s label the number of laps 𝑛 and the distance per lap 𝑑 subscript l. Then we can write our equation a little more concisely as 𝑑 is equal to 𝑛 multiplied by 𝑑 subscript l.
We want to find 𝑛, the number of laps run. So we divide through by 𝑑 subscript l, the distance per lap. Then the 𝑑 l’s on the right-hand side in the numerator and the denominator cancel each other out. Then we have that 𝑛, the number of laps, is equal to 𝑑, the total distance, divided by 𝑑 subscript l, the distance covered in each single lap. We already calculated the total distance 𝑑 as 2200 meters. The question tells us that the distance corresponding to a single lap is 440 meters. This means that we have 𝑑 subscript l is equal to 440 meters.
Substituting in these numbers, we have that the number of laps that the athlete runs is given by 2200 meters divided by 440 meters. This works out to be exactly five laps. We already said that this number of laps run is equivalent to the number of times that the athlete passes the same point on the track. This means that we can check off the second stage of our calculation and that five is the answer to our question.
