2025 VJC P2 Q2

The following diagram shows the dimensions of a trapezoidal prism with fixed volume \(4k\sqrt{3}\) units\(^3\), with variables \(x\) and \(y\).

The top surface of the prism, \(ABCD\), is an isosceles trapezoid with \(AB\) of length \(5x\) units, \(DC\) of length \(3x\) units, \(AD = BC\) and \(\angle ABC = \angle BAD = 60{}^\circ \). The rectangular sides \(ABFE\) and \(BCGF\) are perpendicular to both the top surface \(ABCD\) and the bottom surface \(EFGH\), with \(AE = BF = DH = CG = y\) units.

1. Show that the total external surface area \(A\) of the trapezoidal prism is given by \(A = 8x^2\sqrt{3} + \frac{12k}{x}\).[4]
2. Using differentiation, find the value of \(x\) in terms of \(k\) at which \(A\) is a minimum.[4]
3. It is given instead that the volume of the prism is \(1000\) units\(^3\) and its external surface area is \(800\) units\(^2\). Find the two possible values of \(x\).[2]