Algebra Level 4

Leonardo possess a unique power.He can plant an idea in a persons mind through dreams and can extract information from it.So he plans to extract information from multi- billionaire Chinmay to steal his money. Chinmay has kept his whole wealth amounting to \(1729\) billions in world largest safe which he owns.But this safe has \(50\) digit combination which is known only to Chinmay. So Leonardo uses his skills to find that combination.

To his surprise finding that combination was much harder than he thought.He found \(1\)st digit in his dream.And \(2\)nd in dream within \(1\)st dream.So he got \(50\) digits in dream within dreams(chain of \(50\) dreams).But in first dream \(1\) minute of real world is equal to \(30\) minutes in \(1\)st dreams world.And \(1\) minute of \(1\)st dream's world is equal to \(60\) minutes of \(2\)nd dream's world.Also \(2\)nd dream's \(1\) min is equal to \(120\) minutes of \(3\)rd dream's world and so on following the pattern.

He takes \(10\sqrt { 2 } \) minutes in \(1\)st dream,\(400\) minutes in \(2\)nd dream,\(1000\sqrt { 512 } \) in \(3\)rd dream and so on following a pattern till \(50\)th dream.

So how many minutes of real world he took to find all digits?

Details and Assumptions:

  • Take time of that dream taken by Leonardo to find digit in respective dream.

  • Time taken by him to find digits and rate of time in dream to real world's time also follow patterns.

  • Write answer till 3 decimal places.

  • You can use calculator for some lengthy calculations


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