1.6: Minimise the surface area of a prism

Figure 2 shows a waste paper basket in the shape of a right prism with 5 faces and a cross section that is a trapezium. The top, $\displaystyle EFGH$, of the waste paper basket is open.

The base of the prism $\displaystyle ABCD$ is a rectangle with

The cross sections $\displaystyle HGBA$ and $\displaystyle EFCD$ are such that

The top, $\displaystyle EFGH$, is such that $\displaystyle EH=FG=h\,\mathrm{cm}$.

The volume of the waste paper basket is $\displaystyle 2250\,\mathrm{cm}^3$. The total surface area of the 5 faces 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 gives a minimum value of $\displaystyle S$.

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