All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

(i)
Depth drilled on the 10th day $= 256 +9(-7) = 193$

$T_n \textless 10$

$256 + (n-1)(-7) \textless 10$

$n > 36.1$

Total depth $= S_{37} = \frac{37}{2}[2(256) + 36(-7))] = 4810$

(ii)
$S_n > 0.99 s_{\infty}$

$\frac{256(1-(\frac{8}{9})^n)}{1-\frac{8}{9}} > 0.99 (\frac{256}{1-\frac{8}{9}})$

$(\frac{8}{9})^n \textless 0.01$

$n \mathrm{ln}(\frac{8}{9}) \textless \mathrm{ln} 0.01$

$n > \frac{\mathrm{ln} 0.01}{\mathrm{ln}(\frac{8}{9})} = 39.1$

Thus, it takes 40 days

For (ii), it is worth noting that some students still have no clue how to deal with inequalities when there is $\mathrm{ln}$ involved.