Problem $3(30$ Points$)$:

Assume a file that occupies $20,000$ blocks is sorted using external sort$-$merge, and the number of available memory

blocks is $5.$

Please include all the mathematical steps to compute your final answer for each below question. No point is given

if only the answer is provided.

$(1)$ How many runs are created in the initial pass?

$(2)$ How many merge passes are required? Write the general formula and evaluate it$.$

$(3)$ How many blocks are read or written in total? Include the write of the final output in the calculation.

$(4)$ Assume the time to read or write a block is $15$msec, how much time does the entire sort take?

$(5)$ How many buffer pages are needed to sort the file in just two passes $($one run$-$creating pass and one merge

pass$)?$