The problem between the fox and the duck

In a circular pond, ducks are in the pond, and foxes are on the shore. Ducks can fly away only when they swim to the shore. The fox wants to eat duck. But the fox could not swim and could only run on the shore. The speed of a fox running on the shore is four times that of a duck swimming in the water. Ask the duck how to fly away without being eaten by the fox. Suppose the fox is smart enough.

One feasible idea is that ducks swim in a small circle in the pond, while foxes run in a big circle, but the ducks move farther and farther away. Then when the ducks and foxes are at the longest distance, the duck flew straight to the shore, and the fox had to catch the duck for half a lap, but when it ran to the duck's departure location, the duck had long flown away. Calculate the relationship between the radius of a small circle and the radius of a large circle.

The duck must swim farther and farther down the fox, and the result is:

R/r <1/4 ----------- (1)

Then, when the duck and the fox are at the farthest distance, the duck swam to the shore, while the fox ran half a lap but could not catch the duck, and got another relationship:

(R-R)/V <pI * r/4 V => r-Pi * r/4 ------- (2)

Comprehensive(1) (2)D:

R-Pi * r/4 <r/4 ---------- (3)

This means that the duck chooses an appropriate radius R to meet(3)In this case, the fox had to follow the game on the shore, because the duck would be farther and farther away from the Fox. However, running is useless because the fox will never keep up with the duck speed. When the distance between a duck and a fox isR + RWhen the duck Swam toward the shore, the fox could not catch it. Then the duck escaped successfully.