2023 SAJC P1 Q9

A tentage for an outdoor activity is built by attaching a piece of polyester tent fabric to the tops of four vertical poles \(AE\), \(BF\),\(DG\) and \(CH\), where \(E\), \(F\), \(G\) and \(H\) are positioned at ground level on the \(x\)-\(y\) plane. Coordinates of \(A\), \(B\), \(C\) and \(D\) are as shown in the diagram below, with lengths measured in meters. The length of the pole \(DG\) is \(k\) meters.

1. Find the cartesian equation of the plane \(ABDC\).[3]
2. Find the acute angle that plane \(ABDC\) makes with the horizontal.[2]
3. Find the value of \(k\).[2]
4. Show that the shape of the polyester tent fabric \(ABDC\) is a trapezium. Hence, or otherwise, find its exact area in the form of \({{a}^{\frac{3}{2}}}\) units2 , where \(a\) is an integer to be determined.[5]