2023 ACJC Promo Q11

The diagram shows a roof structure modelled by a prism* \(ABCDEF\) with parallelogram sides. The triangular ends \(ABF\) and \(DCE\) are similar and parallel to the \(xz\) plane. The origin \(O\) is a point on the horizontal ground which is parallel to the \(xy\) plane. The coordinates of point \(B\) and \(M\) are \(\left( 2,-2,11 \right)\) and \(\left( -1.75,-2,8 \right)\) respectively. It is given that plane \(ABCD\) has equation \(48x+5y-60z=-574\)and plane \(BCEF\) has equation \(12x-y+12z=158\).

*While the diagram models a roof structure using a prism, there appears to be a conceptual misunderstanding in the question. A prism, by definition, is a three-dimensional geometric shape with two parallel, congruent bases connected by rectangular or parallelogram faces, which implies a constant cross-sectional area along its length. However, in this case, the structure described seems to have varying cross-sectional areas, which does not conform to the standard geometric properties of a prism. Thus, this structure is **not a prism**.

1. Find the acute angle between plane \(ABCD\) and plane \(BCEF\), giving your answer correct to \(2\) decimal places.[2]
2. Without the use of the calculator, show that the equation of line \(BC\) is \(x=2\), \(\frac{y+2}{12}=z-11\).[2]
3. It is now given that the vertical height of point \(C\) from the horizontal ground is \(11.25\) metres. State the coordinates of point \(C\).[2]
4. Find the acute angle between the line \(BC\) and the horizontal ground, giving your answer correct to \(2\) decimal places.[2]
5. Given that point \(N\) lies on \(BF\) such that \(MN\) is parallel to the \(x\)-axis, find the length of \(MN\).[3]
6. To strengthen the roof structure and minimise material cost, a steel beam \(MW\) is constructed such that point \(W\) lies on plane \(BCEF\) and is closest to point \(M\). Find the length of the steel beam \(MW\).[3]