1.59: Optimising a Cuboid

Figure 3 shows a solid cuboid $\displaystyle ABCDEFGH$.

The volume of the cuboid is $\displaystyle 540\,\mathrm{cm}^3$.

The total surface area of the cuboid is $\displaystyle S\,\mathrm{cm}^2$.

(a) Show that

Given that $\displaystyle x$ can vary,

(b) use calculus to find, to 3 significant figures, the value of $\displaystyle x$ for which $\displaystyle S$ is a minimum.

Justify that this value of $\displaystyle x$ gives a minimum value of $\displaystyle S$.

(c) Find, to 3 significant figures, the minimum value of $\displaystyle S$.