Calculus Level 5

Mayank creates a magical box of side $$s$$. Meanwhile, Akul caught a fly and tied it to an ideal thread. Now, the free end of thread is tied to the center of cube and the setup is left as such.

Why is the box magical?

The box is such that the fly can fly freely inside the cube if $$d<\dfrac { s }{ 2 }$$, where $$d$$ is the distance of the fly from the center.

As soon as its distance from the center exceeds $$\dfrac { s }{ 2 }$$, it will find itself outside the cube with the same distance from the center and can again enter the cube only when its distance from center becomes $$\dfrac { s }{ 2 }$$.

Also as it just enters the cube at d=$$\dfrac { s }{ 2 }$$, It has an equal probability to be at any point at a distance $$\dfrac { s }{ 2 }$$ from the center.

Find the expected distance of the fly from the center of the cube given that the length of thread is $$\dfrac { s }{ \sqrt { 2 } }$$.

If it is equal to $$k\times s$$, find $$\left\lfloor 10^4\times k \right\rfloor$$.

Details and assumptions:

• The fly doesn't feel any tension because of the magical thread.

• The fly can fly!

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