$-$ P$7(10$ pts$)$ Answer the following questions:

B$-$tree Solution for a public library: Suppose we want to design a system to efficiently access the book records in the Omaha public libraries. We assume that we have $70,000,000$ items that each key is $16$ bytes $($representing an ISBN number$),$ and that a record is $600$ bytes $($including authors, publisher, published data etc.$).$ We assume the entire book records do not fit in system's main memory. Suppose one block holds $8,192$ bytes and each branch link is $4$ bytes.

i$)(5$ pts$)$ Find the optimal parameters, M $($branching factor$),$ and L $($number of recorders in a leaf$)$ of the B$-$tree for the system described above.

ii$)(2$ pts$)$ What is the range of the number of leaves in the B$-$tree? $($Justify vour answer$)$

iii$)(3$ pts$)$ What is the maximum height of the B$-$tree constructed using the selected parameter M and L$?($Justify your answer$)$