$($a$)$ Bob and Amy compete in estimating a line with high accuracy. Amy cannot distinguish a line of $8$ cm long within $7\mathrm{.}5$ to $8\mathrm{.}5$ cm $.$ Bob can detect a difference when any line reduces by $\backslash ($ X $\backslash \%\backslash ).$ Suppose their perception follows Weber's law. $\backslash ($ X$=4\backslash )$

$($i$)$ Discuss who can detect a line of $25$ cm long more accurately.

$($ii$)$ Find the range of lengths Bob can distinguish from a line of $\backslash (3$ X $\backslash$mathrm$\{$~cm$\}\backslash )$ long.

$($b$)$ Paul can marginally detect a brightness when he adjusts the surface luminance at near $\backslash (0\mathrm{.}2\backslash$mathrm$\{$~cd$\}/\backslash$mathrm$\{$m$\}^\{2\}\backslash )$ in several times. He can perceive a brightness of $10$ units when the surface luminance is $\backslash (200\backslash$mathrm$\{$~cd$\}/\backslash$mathrm$\{$m$\}^\{2\}\backslash )$ and $15$ units when the surface luminance is increased to $\backslash (5\backslash$boldsymbol$\{$X$\}0\backslash )\backslash (\backslash$mathrm$\{$cd$\}/\backslash$mathrm$\{$m$\}^\{2\}\backslash ).$

$($i$)$ Find the threshold luminance and describe the method used.

$($ii$)$ Evaluate a mathematical expression for his perception on the surface brightness.

$($iii$)$ Calculate the surface luminance if he perceives a brightness of $8$ units.