$($a$)$ A container, of rectangular cross$-$section, is to be designed to maximise its capacity when housed in the void of a parabolic arch bridge.

With coordinate axes, as shown in Fig Q$7($a$),$ the shape of the arch can be represented by the equation $y=4-\frac{1}{4}{x}^2,$ and the unit of length is the metre.

Determine, using differential Calculus, the dimensions of the cross$-$section of the container, represented by the shaded region in Fig Q$7($a$),$ for maximum capacity.