A cuboid has a total surface area of \(\quantity{150} {cm^2}\) and is such that its base is a square of side \(\quantity{x}{cm}\).

Show that the height, \(\quantity{h}{cm}\), of the cuboid is given by \(h=\frac{75-x^2}{2x}\).

Express the volume of the cuboid in terms of \(x\). Hence determine, as \(x\) varies, its maximum volume and show that this volume is a maximum.