A tortoise can run at a speed of 20 cm/s and a hare can run exactly 20 times as fast. In a race, they both start at the same time but the hare stops to rest for 2.00 mins. The tortoise wins by 20.0 cm.

How long does the race take?
What is the length of the race?

The time of the race is the same for both. The distance run by the hare is 20cm less than that of the tortoise. The hare also runs 2min less than the tortoise. Measuring things in cm and sec,

Tortoise: d = 20t
Hare: (d-20) = 400(t-120)

20t-20 = 400t - 48000
47980 = 380t
t = 126.263 sec (time of race)

Checking our answer,
tortoise ran 2525 cm (length of race)
hare ran 2505 cm and lost

To find the length of the race, we can calculate the time it takes for both the tortoise and the hare to reach the finish line.

Let's denote the length of the race as "d" in centimeters.

Given that the tortoise's speed is 20 cm/s, we can determine its time to complete the race using the formula:

Time = Distance / Speed

The time taken by the tortoise to complete the race is:
Time(tortoise) = d / 20 cm/s

The hare's speed is 20 times faster than the tortoise, so its speed is 20 * 20 cm/s = 400 cm/s.

The hare stops to rest for 2.00 mins, which is equal to 2.00 * 60 = 120 seconds. Since the hare stops, the time it takes for the hare to complete the race is:

Time(hare) = (d - 20 cm) / 400 cm/s + 120 s

The tortoise wins by 20.0 cm, so the time taken by the tortoise to reach the finish line is the same as the hare's time minus the resting time:

Time(tortoise) = Time(hare) - 120 s

Next, we can equate the equations for the tortoise and hare's times:

d / 20 cm/s = (d - 20 cm) / 400 cm/s + 120 s

To solve this equation, we can multiply both sides by 20 cm/s and 400 cm/s to eliminate the fraction:

400 * d = 20 * (d - 20) + 120 * 400

Now, we can solve for "d":

400d = 20d - 400 + 48000
380d = 47600
d ≈ 125.26 cm

Therefore, the length of the race is approximately 125.26 cm.

To find the total time taken for the race, we can substitute the value of "d" into one of the time equations. Let's use the tortoise's time equation:

Time(tortoise) = d / 20 cm/s
Time(tortoise) = 125.26 cm / 20 cm/s
Time(tortoise) ≈ 6.26 s

Hence, the race takes approximately 6.26 seconds.

To calculate the time it takes for the race, we need to determine when the tortoise and hare reach the finish line.

First, let's find the time it takes for the tortoise to finish the race:
Distance = Speed * Time
The tortoise's speed is 20 cm/s, and the distance it travels is the length of the race. Let's call the length of the race "L" cm.
So the time it takes for the tortoise to finish the race is:
Time(tortoise) = Distance/Speed = L/20 seconds

Now let's find the time it takes for the hare to finish the race, taking into account that it rests for 2.00 minutes (or 120 seconds):
Time(hare) = (L/20) + 120 seconds

Since we know that the tortoise wins by 20.0 cm, we can set up the equation:
L/20 - L/20 = 20 cm
1 cm = 20 cm
L = 400 cm

So the length of the race is 400 cm.

Now, let's substitute L into the equation for the time it takes for the tortoise to finish the race:
Time(tortoise) = 400/20 seconds = 20 seconds

Now we can calculate the time it takes for the hare to finish the race:
Time(hare) = (400/20) + 120 seconds = 20 + 120 seconds = 140 seconds

Therefore, the race takes 140 seconds (or 2 minutes and 20 seconds) to complete.