Higher November 2021 Paper 1 Q12
12 The diagram shows a vertical cliff with a vertical radio mast on top of the cliff and a buoy in the sea.
Diagram NOT accurately drawn
The height of the cliff is 100 metres.
The buoy is at the point \(B\) that is \(d\) metres from the base of the cliff.
The angle of elevation from \(B\) to the top of the cliff is 20°
(a) Calculate the value of \(d\).
Give your answer correct to 3 significant figures. (3)
Give your answer correct to 3 significant figures. (3)
The point \(A\) at the top of the radio mast is vertically above the top of the cliff.
The angle of elevation from \(B\) to \(A\) is 25°
(b) Calculate the height of the radio mast.
Give your answer correct to 3 significant figures. (3)
Give your answer correct to 3 significant figures. (3)
| Scheme | Marks |
|---|---|
\(\tan 20 = \dfrac{100}{d}\) oe or \(\tan(90 - 20) = \dfrac{d}{100}\) oe or \(\dfrac{d}{\sin(90 - 20)} = \dfrac{100}{\sin 20}\) oe | M1 |
\((d =)\ \dfrac{100}{\tan 20}\ (= 274.747\ldots)\) or \((d =)\ 100 \times \tan(90 - 20)\ (= 274.747\ldots)\) or \((d =)\ \dfrac{100}{\sin 20} \times \sin(90 - 20)\ (= 274.747\ldots)\) | M1 |
| Correct answer scores full marks (unless from obvious incorrect working) Answer: 275 | A1 |
| (3) |
Notes
A1: awrt 275
| Scheme | Marks |
|---|---|
\(\tan 25 = \dfrac{100 + h}{275}\) oe or \(\tan 25 = \dfrac{y}{275}\) oe or \(275 \times \tan 25\ (= 128\ldots)\) or \(\tan(90 - 25) = \dfrac{275}{100 + h}\) oe or \(\tan(90 - 25) = \dfrac{275}{y}\) oe or \(\dfrac{100 + h}{\sin 25} = \dfrac{275}{\sin(90 - 25)}\) or 128.1 – 128.2 (\(y\) is the height of cliff and radio mast) | M1 |
\((h =)\ 275 \times \tan 25 - 100 = 28.1169\ldots\) or \((h =)\ \dfrac{275}{\tan(90 - 25)} - 100\ (= 28.1169\ldots)\) or \((h =)\ \dfrac{275}{\sin(90 - 25)} \times \sin 25 - 100\ (= 28.1169\ldots)\) | M1 |
| Correct answer scores full marks (unless from obvious incorrect working) Answer: 28.1 | A1 |
| (3) | |
| (6 marks) |
Notes
M1: ft part (a) Allow
\((hyp =)\ \sqrt{100^2 + 275^2}\ (= \sqrt{85486.321} = 292.380)\)
or
\((hyp =)\ \dfrac{100}{\sin 20} \times \sin 90\ (= 292.380)\)
M1: ft part (a)
\((h =)\ \dfrac{\text{``}{292.380}\text{''}}{\sin(90 - 25)} \times \sin(25 - 20)\)
(= 28.1169…)
A1: Accept answers in the range 28 – 28.2