A hare and a tortoise compete in a race over a course 1.00 km long. The tortoise crawls straight and steadily at its maximum speed of 0.200 m/s toward the finish line.The hare runs at its maximum speed of 8.00 m/s toward the goal for 0.800 km and then stops to tease the tortoise.How close to the goal can the hare let the tortoise approach before resuming the race, which the tortoise wins in a photo finish? Assume that, when moving, both animals move steadily at their respective maximum speeds.

To solve the problem of the hare and the tortoise racing over a 1.00 km course, we need to analyze their speeds and the time each takes to reach the finish line. The hare runs much faster than the tortoise but stops to tease it, which is crucial for determining how close the tortoise can get to the finish line before the hare resumes running. Let's break this down step by step.

Understanding the Speeds

The speeds of the two competitors are:

• The tortoise: 0.200 m/s
• The hare: 8.00 m/s

Calculating Time for the Tortoise

First, we need to find out how long it takes the tortoise to complete the entire 1.00 km (or 1000 meters) course:

Time = Distance / Speed

For the tortoise:

Time_tortoise = 1000 m / 0.200 m/s = 5000 seconds

Calculating the Hare's Initial Run

Next, let's determine how far the hare can run before it stops to tease the tortoise. The hare runs for 0.800 km (or 800 meters):

Time_hare = Distance / Speed

For the hare:

Time_hare = 800 m / 8.00 m/s = 100 seconds

Distance Covered by the Tortoise During the Hare's Run

While the hare is running for 100 seconds, we can calculate how far the tortoise has crawled:

Distance_tortoise = Speed × Time

Distance_tortoise = 0.200 m/s × 100 s = 20 meters

Position of the Hare and Tortoise After 100 Seconds

After 100 seconds:

• The hare has covered 800 meters and is at the 800-meter mark.
• The tortoise has covered 20 meters and is at the 20-meter mark.

Teasing and Resuming the Race

Now, the hare stops at the 800-meter mark to tease the tortoise. At this point, the tortoise is 980 meters away from the finish line (1000 m - 20 m). The hare will resume running once the tortoise gets close enough to finish the race at the same time.

Time for the Tortoise to Reach the Finish Line

To find out how long it takes the tortoise to reach the finish line from its current position:

Remaining distance for the tortoise = 1000 m - 20 m = 980 m

Time_{tortoise to finish} = 980 m / 0.200 m/s = 4900 seconds

How Far Can the Hare Let the Tortoise Approach?

During the 4900 seconds that the tortoise takes to finish, the hare will also be running. The hare resumes running after 100 seconds, so it has 4900 seconds - 100 seconds = 4800 seconds to run.

Distance covered by the hare in 4800 seconds:

Distance_hare = Speed × Time = 8.00 m/s × 4800 s = 38400 meters

Since the hare is already at the 800-meter mark, it can run an additional 38400 meters, which is far beyond the finish line. Thus, the hare can let the tortoise approach very close to the finish line before it resumes running.

Final Calculation

To ensure a photo finish, the hare must time its run so that it reaches the finish line at the same time as the tortoise. The hare can let the tortoise get as close as:

Distance_tortoise = 1000 m - Distance_hare (which is 800 m) = 200 m

Thus, the hare can let the tortoise approach to within 200 meters of the finish line before it resumes running, ensuring a close finish.

In summary, the hare can let the tortoise get to the 800-meter mark before it resumes running, allowing for a thrilling photo finish!