Higher June 2018 Paper 3 Q25

25 \(ABCDEF\) is a triangular prism which represents part of a hill.

\(ABCF\) is the horizontal rectangular base.
\(D\) is vertically above \(C\).

(a) Work out the height \(CD\). [2 marks]

(b) Jamil walks in a straight line from \(A\) to \(D\).

Work out the size of angle \(DAC\).

You must show your working. [4 marks]

Mark scheme (a)

Answer	Mark	Comments
\(\tan 6 = \dfrac{CD}{500}\) or \(500 \times \tan 6\)	M1	oe any letter \(\dfrac{CD}{\sin 6} = \dfrac{500}{\sin 84}\)
[52.5, 52.6] or 53	A1	May be on diagram

Check diagram for angle

Mark scheme (b)

Answer	Mark	Comments
Alternative method 1
\(500^2 + 400^2\) or \(250\,000 + 160\,000\) or \(410\,000\)	M1	oe
\(\sqrt{\text{their } 410\,000}\) or \(\sqrt{500^2 + 400^2}\) or 640.(3…)	M1dep	\(AC\)
\(\tan x = \dfrac{[52.5, 52.6] \text{ or } 53}{\text{their } 640.(3..)}\)	M1dep	oe any letter
[4.6, 4.75] from correct working	A1	accept 5 with correct working seen
Alternative method 2
\(\dfrac{500}{\cos 6}\) or [502.7, 502.8]	M1	oe \(BD\)
\(\sqrt{\left(\dfrac{500}{\cos 6}\right)^2 + 400^2}\) or [642.4, 642.5]	M1dep	\(AD\)
\(\sin x = \dfrac{[52.5, 52.6] \text{ or } 53}{\text{their } [642.4, 642.5]}\)	M1dep	oe any letter
[4.6, 4.75] from correct working	A1	accept 5 with correct working seen
Alternative method 3
\(500^2 + 400^2\) or \(250\,000 + 160\,000\) or \(410\,000\) or \(\dfrac{500}{\cos 6}\) or [502.7, 502.8]	M1	oe \(BD\)
\(\sqrt{\text{their } 410\,000}\) or \(\sqrt{500^2 + 400^2}\) or 640.(3…) or \(\sqrt{\left(\dfrac{500}{\cos 6}\right)^2 + 400^2}\) or [642.4, 642.5]	M1dep	\(AC\) \(AD\)
\(\cos x = \dfrac{\text{their } 640.(3...)}{\text{their } [642.4, 642.5]}\)	M1dep	oe any letter
[4.6, 4.75] from correct working	A1	accept 5 with correct working seen

Check diagram for lengths
Beware \(\sin x = \dfrac{52.6}{640.(3...)}\) leads to [4.6, 4.75]	M1M1M0A0