400 on top of 4 100s

Level pending

Four spheres of radii 100 each rest on a horizontal plane H such that their centers form a square of side 200. A sphere of radius 400 rests over them.

Let d be the largest possible distance of a point on the larger sphere from the plane H. What is integral part of d?


The answer is 979.

Piyushkumar Palan by Piyushkumar Palan , 31, India

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1 solution

Aditya Raut
Jun 12, 2014

Imagine the pyramid made by the 4 centres and the centre of big sphere.

It has height 50 0 2 − ( 100 2 ) 2 = 250000 − 20000 = 230000 \sqrt{500^2 - (100\sqrt{2})^2}=\sqrt{250000-20000}=\sqrt{230000}

Required height is radius 1 + height of pyramid + radius 2 .

And thus required answer is 100 + 400 + ⌊ 230000 ⌋ = 479 + 500 = 979 100 + 400 +\lfloor \sqrt{230000} \rfloor = 479+500 = \boxed{979}

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