Q$8.$ A robot arm with a rapid laser scanner is doing a quick quality check on holes drilled in a $15xx{10}^{\text{'}\text{'}}$ rectangular plate. The centers of the holes in the plate describe the path the arm needs to take, and the hole centers are located on a Cartesian coordinate system $($with the origin at the bottom left corner of the plate$)$ given by the specifications in table below

$\backslash$table$[[x($in$.),y=f(x)($in$.)],[2\mathrm{.}00,7\mathrm{.}2],[4\mathrm{.}25,7\mathrm{.}1],[5\mathrm{.}25,6\mathrm{.}0],[7\mathrm{.}81,5\mathrm{.}0],[9\mathrm{.}20,3\mathrm{.}5],[10\mathrm{.}60,5\mathrm{.}0]]$

If the laser is traversing from $x=2\mathrm{.}00$ to $x=4\mathrm{.}25$ to $x=5\mathrm{.}25$ in a quadratic path, what is the value of $y$ at $x=4\mathrm{.}00$ using a second order Lagrange polynomial? Find the absolute relative approximate error for the second order polynomial approximation if the exact has been considered $7\mathrm{.}1111$ in$.$