Higher June 2022 Paper 3 Q18
18 Here is a prism \(ABCDSPQR\).
The base \(ABCD\) of the prism is a square of side 14 cm
\(T\) is the point on \(BC\) such that \(BT : TC = 4 : 3\)
The cross section of the prism is in the shape of a trapezium of area 147 cm2
\(CR = 12\) cm
Find the size of the angle between the line \(ST\) and the base \(ABCD\).
Give your answer correct to 1 decimal place. (5)
| Answer | Mark | Mark scheme |
|---|---|---|
| 30.6 | P1 | for process to find \(TC\), eg (\(TC =\)) \(14 \times \dfrac{3}{3 + 4}\ (= 6)\) |
| P1 | for process to find \(TD\), eg (\(TD =\)) \(\sqrt{14^2 + \text{``}6\text{''}^2}\) or \(\sqrt{232}\) or \(2\sqrt{58}\ (= 15.2\ldots)\) | |
| P1 | for process to find \(SD\), using area of a trapezium, \(147 = 0.5 \times (SD + 12) \times 14\), or \(SD = 9\) | |
| P1 | for \(\tan^{-1}\left(\dfrac{\text{``}9\text{''}}{\text{``}15.2\ldots\text{''}}\right)\) | |
| A1 | for answer in the range 30.4 to 30.7 |
Additional guidance
Lengths of \(TC\), \(TD\), \(SD\) may be seen on the diagram
A complete set of processes to find the angle is needed where an alternative route is involved with more than one stage in the working
If an answer is given in the range in working and then rounded incorrectly award full marks.