# Water flowing from a sphere

Calculus Level 5

A hollow sphere of mass $$m$$ and radius $$1$$ metre is placed with it's centre at origin in $$xyz$$-space. It is filled completely with water of mass $$m$$. So, now the total mass of system is $$2m$$.

A small hole is made at $$t=0$$ at the bottom and water starts flowing out of the sphere with a constant volume flow rate.

If the maximum distance of centre of mass of the system from origin during the time $$t=0$$ to the time water completely flows out from the sphere is $${a}^{\frac{1}{n}}-{b}^{\frac{1}{n}}$$ meters, then what is the value of $$a+b+n?$$

Details and assumptions

• $$a,b,n$$ are distinct integers such that $$0<a,b,n<20$$.

• While calculating centre of mass of the system don't consider the water that flown out from the sphere. Only consider the water that is present inside the sphere.

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