Required Mathematical Intelligence

Chapter 120 Can the mouse escape the cat?
A mouse jumps into a circular lake with a radius of R in order to avoid being chased by a cat. The cat can’t swim, so it can only chase along the shore of the lake, and always tries to make itself the closest to the mouse (that is, the cat always tries to keep itself in the The point where the mouse is closest to the shore), assuming that the maximum speed of the cat on land is 4 times the maximum speed of the mouse swimming in the lake, ask whether the mouse can get rid of the pursuit of the cat? (If the cat is not in the position where the mouse landed when the mouse landed, it is considered that the mouse has escaped from the cat's pursuit)
[Answer: Make a circle with the center of the lake as the center and R/4 as the radius.If the mouse swims along this circle, then the mouse in the water and the cat on the shore have the same angular velocity. If the radius of the mouse swimming is slightly smaller than R/4, set R', it will have a greater angular velocity than the cat. If the mouse The swimming time is long enough to be 180 degrees ahead of the cat, that is, the mouse is slightly less than R' to the left of the origin, and the cat is at R to the right of the origin.So now the mouse will swim R-R' and the cat will run 3.14R.if only:
4(R-R')<3.14R...(1) That is, the cat has not run to the landing place when the mouse lands, and R'<R/4, the mouse can escape from the cat's pursuit. In fact, this is completely OK.

Solve formula (1) to get:
R'＞0.215R
Intersect with R' < 0.25R, so mice can escape. ]
(End of this chapter)