Unleash Me!
On the big green lawn, Lucky the dog was tethered with a 3-meter rope, tied firmly at stake A, while the 3 stakes A, B, C formed an equilateral triangle of 1 meter apart from one another, as shown on the left.
Starting from point A, Lucky eagerly struggled to run away, pulling the rope tight at all time, before encircling around stake B in a clockwise fashion. Once the rope made a linear alignment past the stake, the naughty dog could run more freely with a longer radius. Nevertheless, the rope would eventually wind around each stake once more, and the ends of the rope would consequently meet up at point A again.
Finally, after completing the clockwise trip, Lucky gave up and took a rest at point A, as shown on the right.
What was the total distance in meters that Lucky had traveled? If the total distance is meters, submit your answer as .
Assume that the distance off the ground was insignificant and Lucky's path made a perfectly circular arc.
The answer is 9.
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From the first round around stake B, the radius is 1 meter, which can go for 120 degrees, so the total arc is 1/3(2 π ) = 2 π /3.
From the first round around stake C, the radius is 2 meters, which can go for 120 degrees, so the total arc is 1/3(4 π ) = 4 π /3.
From the first round around stake A, the radius is 3 meters, which can go for 300 degrees, so the total arc is 5/6(6 π ) = 5 π .
From the second round around stake B, the radius is 2 meters, which can go for 120 degrees, so the total arc is 1/3(4 π ) = 4 π /3.
From the first round around stake C, the radius is 1 meter, which can go for 120 degrees, so the total arc is 1/3(2 π ) = 2 π /3.
Therefore, the total distance = 2 π /3 + 4 π /3 + 5 π + 4 π /3 + 2 π /3 = 9 π .