B$2.$

$($a$)$ A rectangular metal sheet of $600mm$ by $400mm$ has four equal squares cut out at the corners. The sides are then turned up to form a rectangular box. The length of the side of each cut$-$away square by $xmm$ is shown in Figure B$2.$

$($i$)$ Find the length of the sides of the squares cut out so that the volume of the box is maximum.

$(12$ marks$)$

$($ii$)$ Find the corresponding maximum volume.

$(3$ marks$)$

$($b$)$ Find the equation of the tangent to the curve $y=x^2-3x,$ at the point $(4,4).$

$(5$ marks$)$