Algebra homework help

Question

Your are Olympic runner. You have just qualified to be in finals of the 1500-meter race. The track is 400 meters in an oval shape. The race is three and three-fourths laps around the track.

The favorite to win the race is Kenyan who holds the current best time, which is 3 minutes 29.4 seconds. The Kenyan runs a very steady race. Each of the Kenyan’s lap times(400 meters) is within a second of each other.

You run a completely different type of race. You have a vary strong kick, which means you usually lag behind for the first three laps to save energy, and then when the leader has 300 meters to go you pour it on to win the tape. You like to save energy in the first three laps, but you don’t want to be more than 50 meters behind when you start your kick to finish line.

Determine your strategy to win the race. What is the average speed you need to run during the first part of the race? What is the average speed you need to run during your kick to win the race? How might your race change if the Kenyan runs two seconds faster?

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How do you do this?

Answer 1

It sounds like you should split the race into two sections: the first 1150 meters and the last 350 meters.

Calculate the Kenyan's average speed over the whole 1500. (1500 / 209.4 seconds). That's his meters per second. (looks to be about 7.2, but most teachers don't want you to round unless it's part of an answer. If you do, then use whichever rounding rules your teacher suggests--either by significant digits or to a specific decimal point.)

Use that to figure out what time he hits 1200 m (which is also the time you will hit 1150 meters). Looks like 167.52 seconds.

So, up to that point, you'll want to cover at least 1150m in 167.52 seconds, which is about 6.86 m/s.

So s₁ > 6.86 m/s

Since we know the Kenyan will cross the tape at 209.4s, and you've spent 167.52s already, you've got 41.88s to go in the race. And you need to cover 350m in that time. Time to kick up your speed.

350/41.88 is about 8.36 meters per second. So s₂ > 8.36 m/s

For the final part of the question, replace 209.4 with 207.4 and run everything through your calculator again.

Answer 2

Let's break this down and instead of looking at the problem in its whole, look at each piece separately. The Kenyan completes his lap in 209.4 seconds, which allows us to conclude his average lap speed:

(209.4 seconds) x (400 meters/1500 meters) = 55.84 seconds per lap (400 m) = 7.16332 m/s

Now that we know the Kenyan's speed, we need to find the speed of yourself so that you are 50 m behind the Kenyan after the third lap. In this case, you would have ran 1150 m in the time that the Kenyan ran 1200 m. This is how you can find your speed:

(1150 m/(55.84 s x 3 laps)) = 6.864851958 m/s = 58.26780062 seconds per lap (first 1150 m)

Now, to win the race, we have to run 350 m in the time that is left in order to beat the Kenyan:

209.4 s - (55.84 s x 3 laps) = 41.88 seconds left

Now that we have the time that is left, our speed is found by dividing the meters to run by the time left:

350 m/41.88 s = 8.357211079 m/s during the last part (last 350 m)

If the Kenyan were to go two seconds faster, the exact same calculations would be done with 207.4 seconds instead of 209.4 seconds:

Average Kenyan Speed = 207.4 s x (4/15) = 55.30667 seconds per lap = 7.232401157 m/s

Our Speed = (1150 m/(55.30667 x 3 laps) = 6.931050691 m/s = 57.71120783 seconds per lap (first 1150 m)

Time left = 207.4 s - (55.30667 s x 3 laps) = 41.47999 seconds left

Speed during last part = 350 m/41.47999 s = 8.437803384 m/s (last 350 m)

*Be sure to round all answers if significant digits are required*