$(15$ points$)$ Using two relations R and S $,$ with the join condition $Ri=Sj,$ and given the following information: $($Show all your work$)$

M pages in R with pR tuples per page

$,N$ pages in S with pS tuples per page

R contains:

$900$ pages

$30$ tuples per page

S contains:

$1000$ pages

$10$ tuples per page

Buffer size is $102($inclusive of $2$ additional buffers$)$

$($a$)$ Compute the I$/$O cost for R and S using a block nested loop join for both:

$R|||><mi>|</mi><mi>S</mi><mi>|</mi><mi>|</mi>$

$S|||><mi>|</mi><mi>R</mi><mi>|</mi><mi>|</mi>$

$($b$)$ Compute the I$/$O cost for R and S using a hash join.