An Olympic athlete is helping her niece train. The athlete can run short distances at nine meters per second, and her niece can run short distances at seven meters per second. The athlete gives her niece a two-second head start in a sprint race, and her niece runs for 𝑡 seconds. Write an inequality for the values of 𝑡 when the athlete is behind her niece. Assume that they run at steady speeds the entire race.
There are several ways of approaching this problem. The first one involves creating a table of distances. We can create a table with three rows: the time in seconds, the distance covered by the niece in meters, and the distance covered by the athlete in meters. Let’s initially consider the time period from one second to 10 seconds. We are told that the niece can run short distances at seven meters per second. This means that after one second, she will have run seven meters. As she is running at a steady speed, after two seconds, she will have run 14 meters. We can continue this pattern by adding seven meters for each second.
The athlete gave her niece a two-second head start. This means that for the first two seconds, she will not have covered any distance. As the athlete runs at nine meters per second, we can then increase the distance by nine meters for each second. After three seconds, she will have covered nine meters. After four seconds, she will have covered 18 meters. This pattern will continue as shown. We can see that from one second to eight seconds, the niece is ahead as they have covered a greater distance. After 10 seconds, the athlete is ahead. They have covered 72 meters, whereas the niece covered 70 meters. However, after nine seconds, they have both run the same distance of 63 meters. We can therefore conclude that the athlete is behind her niece when 𝑡 is less than nine seconds. The correct inequality is 𝑡 is less than nine.
An alternative method to solve this problem would be to use our speed–distance–time triangle. This tells us that distance is equal to speed multiplied by time. As the niece ran at a speed of seven meters per second for 𝑡 seconds, her distance will be equal to seven multiplied by 𝑡. This can be written as seven 𝑡. The athlete ran at a speed of nine meters per second. As she started running two seconds after her niece, the time she will be running for is 𝑡 minus two. We can calculate the distance by multiplying nine by 𝑡 minus two.
We can distribute the parentheses or expand the brackets here by multiplying nine by 𝑡 and nine by negative two. This gives us a distance of nine 𝑡 minus 18. We need to calculate the time when the athlete is behind her niece. This will occur when she has covered a smaller distance. We can write this as the inequality nine 𝑡 minus 18 is less than seven 𝑡. By adding 18 and subtracting seven 𝑡 from both sides of this inequality, we get nine 𝑡 minus seven 𝑡 is less than 18. The left-hand side simplifies to two 𝑡. We can divide both sides of this inequality by two, leaving us with the answer of 𝑡 is less than nine. The athlete is behind her niece for any time up to nine seconds.