Fig. 1 Fig. 2 Fig. 3 Fig. 1 shows a piece of card, $ABC$, in the form of an equilateral triangle o

Question

Fig. 1

Fig. 2

Fig. 3

Fig. 1 shows a piece of card, $ABC$, in the form of an equilateral triangle of side $a$. A kite shape is cut from each corner, to give the shape shown in Fig. 2. The remaining card shown in Fig. 2 is folded along the dotted lines, to form the open triangular prism of height $x$ shown in Fig. 3.

Show that the volume $V$ of the prism is given by $V=\frac14x\sqrt3(a-2x\sqrt3)^2$.[3]
Use differentiation to find, in terms of $a$, the maximum value of $V$, proving that it is a maximum.[6]

Accepted Answer

(i) $V=\frac14x\sqrt3(a-2x\sqrt3)^2$ (shown). (ii) $V_{\max}=a^3/54$, at $x=a/(6\sqrt3)$.

N2013 P2 Q2

N2013 P2 Q2

Solution:

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