A tortoise and a hare are having a 1000-meter race. The tortoise runs the race at a constant speed of 2.30 cm/s. The hare moves at an average speed of 1.50 m/s for 10.0 minutes and then decides to take a nap. After waking up from the nap, the hare recognizes that the tortoise is about to cross the finish line and immediately accelerates from rest with a constant acceleration of 0.500 m/s/s for the remaining distance of the race. If the tortoise wins by a hair (no pun intended), then what is the time in hours that the hare napped?

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To find the time in hours that the hare napped, we need to calculate the time it took for the tortoise to finish the race and then subtract it from the total time the hare took.

First, let's find the time it took for the tortoise to finish the race. We know that the tortoise runs at a constant speed of 2.30 cm/s, and the distance of the race is 1000 meters. We need to convert the speed to meters per second to match the distance unit:

2.30 cm/s = 0.023 m/s

Now we can calculate the time it took for the tortoise to finish the race:

Time = Distance / Speed
Time = 1000 m / 0.023 m/s

Calculating this gives us:

Time = 43478.26 seconds

Next, let's calculate the time it took for the hare to reach the point where it decided to take a nap. The hare moved at an average speed of 1.50 m/s for 10.0 minutes, which is equivalent to 600 seconds:

Distance = Speed * Time
Distance = 1.50 m/s * 600 s
Distance = 900 meters

So, the hare covered a distance of 900 meters in 600 seconds.

Now, let's calculate the remaining distance the hare needs to cover after waking up from the nap:

Remaining distance = Total distance - Distance covered before the nap
Remaining distance = 1000 m - 900 m
Remaining distance = 100 meters

Now, let's calculate the time it took for the hare to cover the remaining distance with a constant acceleration of 0.500 m/s/s. We can use the equation of motion:

Distance = Initial velocity * Time + 0.5 * Acceleration * Time^2

Since the hare starts from rest (initial velocity = 0), the equation simplifies to:

Distance = 0.5 * Acceleration * Time^2

Plugging in the values:

100 m = 0.5 * 0.500 m/s^2 * Time^2

Simplifying the equation gives us:

Time^2 = (100 m * 2) / 0.500 m/s^2
Time^2 = 400 s^2
Time = √(400 s^2)
Time = 20 s

Now, let's calculate the total time the hare took for the race:

Total time = Time before nap + Time after nap
Total time = 600 s + 20 s
Total time = 620 s

Finally, let's convert the total time to hours:

Total time in hours = Total time in seconds / 3600 s/hour
Total time in hours = 620 s / 3600 s/hour

Calculating this gives us:

Total time in hours = 0.172 hours

Therefore, the hare napped for approximately 0.172 hours or 10.32 minutes.