1.38: Past-paper question 8

Figure 4 shows a solid right triangular prism $\displaystyle ABCDEF$

The cross section of the prism is an isosceles triangle.

• $\displaystyle \angle DEC = \angle AFB = 90^\circ$

• $\displaystyle AB = DC = x$ cm

• $\displaystyle AD = BC = FE = y$ cm

• $\displaystyle AF = BF = DE = CE$

The triangular faces of the prism are vertical and the edges $\displaystyle AD$, $\displaystyle BC$ and $\displaystyle FE$ are horizontal.

The volume of the prism is $\displaystyle 3.6 \mathrm{cm}^3$

The total external surface area of the prism is $\displaystyle S \mathrm{cm}^2$

(a) Show that $\displaystyle S$ satisfies the equation

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) Hence find, to 2 significant figures, the minimum value of $\displaystyle S$