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 \(h=\frac{75-x^2}{2x}\).

How can we find the surface area of a cuboid from its sides?

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.

Can we write the volume of the cuboid in terms of \(x\) and \(h\)? Can we now substitute for \(h\)?

How can we find the maximum value?

This applet shows how the shape of the cuboid changes as we vary \(x\). The surface area is fixed and the length of the red line is proportional to the volume.