A tortoise can run with a speed of 11.5 cm/s, and a hare can run 21.4 times as fast. In a race, they both start at the same time, but the hare stops to rest for 126 seconds. The tortoise wins by a shell (18.3 cm). How long does the race take? What is the length of the race?

if the race is over distance d cm, then at the end, the tortoise has gone d, and the hare has only gone d-18.3

If the tortoise goes at speed 11.5 cm/s, the hare runs at 21.4*11.5=246.1

Now, the hare's running time is 126 s less than the tortoise's.

Since the total times are equal, and time = distance/speed, we have

d/11.4 = (d-18.3)/246.1 + 126
d = 1507.06 cm

check:

1507.06/11.5 = 132.2 sec
1488.76/246.1 = 6.1 sec + 126 = 132.1

To find the length of the race, we need to determine the time it takes for the tortoise to complete the race. Since the tortoise wins by a shell length of 18.3 cm, we can calculate the time it takes for the tortoise to travel this distance.

The speed of the tortoise is given as 11.5 cm/s, so we can use the formula:

Time = Distance / Speed

Substituting the values, we have:

Time = 18.3 cm / 11.5 cm/s

Calculating this gives us:

Time = 1.59 seconds (rounded to two decimal places)

Now, let's determine the time it takes for the hare to travel the same distance. The hare is 21.4 times faster than the tortoise, so its speed is:

Speed of Hare = Speed of Tortoise * 21.4 = 11.5 cm/s * 21.4 = 246.1 cm/s

However, the hare stops to rest for 126 seconds. So, we need to deduct this resting time from the overall race time of the hare:

Total Time of Hare = Time of Rest + Time to Cover Distance
= 126 seconds + (18.3 cm / 246.1 cm/s)
= 126 seconds + 0.074 seconds (rounded to three decimal places)

Calculating this gives us:

Total Time of Hare = 126.074 seconds (rounded to three decimal places)

As both the tortoise and the hare start at the same time, the race takes the same time as the hare:

Race Time = Total Time of Hare = 126.074 seconds (rounded to three decimal places)

To find the length of the race, we multiply the race time by the hare's speed:

Length of Race = Race Time * Speed of Hare
= 126.074 seconds * 246.1 cm/s

Calculating this gives us:

Length of Race = 31,031.6734 cm (rounded to two decimal places)

So, the race takes approximately 126.074 seconds, and the length of the race is around 31,031.67 cm.