<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1781766633273-id5gp2n3uel.jpg" style="max-width: min(500px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Fig. 1 shows a piece of card in the form of a square of side ${2\sqrt{2}\ a}$. Four isosceles triangles of height ${\sqrt{2}\ x}$ are cut from the card to give the shape shown in Fig. 2. The remaining card shown in Fig. 2 is folded along the dotted lines, to form the square pyramid as shown in Fig. 3. If&nbsp;${V}$ is the volume of the square pyramid, show that&nbsp;${V = k(a - x)^2\ \sqrt{ax}}$, where&nbsp;${k}$ is an exact real constant to be determined. Hence, using differentiation, find the maximum value of ${V}$ in terms of ${a}$.<span class="inline-marks">[9]</span></p>
<p>[Volume of a right square pyramid is $\frac{1}{3}$(base area)(perpendicular height).]</p>

Fig. 1 shows a piece of card in the form of a square of side ${2\sqrt{2}\ a}$. Four isosceles triangles of height ${\sqrt{2}\ x}$ are cut from the card to give the shape shown in Fig. 2. The remaining card shown in Fig. 2 is folded along the dotted lines, to form the square pyramid as shown in Fig. 3. If ${V}$ is the volume of the square pyramid, show that ${V = k(a - x)^2\ \sqrt{ax}}$, where ${k}$ is an exact real constant to be determined. Hence, using differentiation, find the maximum value of ${V}$ in terms of ${a}$.

[Volume of a right square pyramid is $\frac{1}{3}$(base area)(perpendicular height).]

Applications of Differentiation

Applications of Differentiation