The answer
(a) shown (\(x = 5.7\))
(b) shown (\(OC = 38.5\) m)
(c) Yes, total \(\approx 2082\) kg \(< 2100\) kg
O-Level E-Math 2021 Paper 2 Question 10 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 10 of the O-Level E-Math 2021 Paper 2. It tests gradient to angle (tan), in the Pythagoras & trigonometry / Mensuration area. It is worth 10 marks: 1 + 3 + 6. It is a worded / diagram-based question, so open your Ten-Year Series (TYS) or the official paper at this question, then follow our full worked solution below.
(a) \(\tan x^{\circ} = \dfrac{\text{vertical}}{\text{horizontal}} = \dfrac{1}{10} = 0.1\), so \(x = \tan^{-1}(0.1) = 5.71^{\circ} = 5.7^{\circ}\) (1 d.p.). (shown)
(b) Take \(B\) at the origin and the top of the mast \(O = (0, 35)\). The deck runs at \(5.7^{\circ}\) to the horizontal, so \(C\) (20 m along the deck from \(B\), towards \(D\)) is at \((20\cos 5.7^{\circ},\ 20\sin 5.7^{\circ}) = (19.90,\ 1.99)\). Then \(OC = \sqrt{19.90^2 + (1.99 - 35)^2} = \sqrt{1486} = 38.55 = 38.5\) m (1 d.p.). (shown)
(c) Cable lengths: \(OA = 48.3\), \(OC = 38.55\), and \(OD = \sqrt{(40\cos5.7^{\circ})^2 + (40\sin5.7^{\circ} - 35)^2} = 50.47\) m. Unit masses: 64 mm \(\to\) FLC 64 \(= 23.2\) kg/m; 44 mm \(\to\) FLC 44 \(= 10.8\) kg/m. - \(OA\): \(48.3 \times 23.2 = 1120.6\) kg. - \(OC\): \(38.55 \times 10.8 = 416.3\) kg. - \(OD\): \(50.47 \times 10.8 = 545.0\) kg. Total \(= 1120.6 + 416.3 + 545.0 = 2082\) kg. Since \(2082\) kg \(< 2100\) kg \((= 2.1\) tonnes\()\), the cables meet the engineer's specification (with about 18 kg to spare).
Answer: (a) shown (\(x = 5.7\))
(b) shown (\(OC = 38.5\) m)
(c) Yes, total \(\approx 2082\) kg \(< 2100\) kg
Same structure, different numbers
Swap the constants, dress a quadratic as a length, hide a derivative inside an integral, and a student sees a brand new problem. The structure underneath is the same, and so is the method. Once a student can name the structure, a whole row of questions that look different start to open the same way.
That is where marks really leak: in choosing the method, not in the algebra that follows. We call it Lock and Key, name the lock, then the key follows.
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Genius Plus Academy · O-Level & IP Mathematics
Our O-Level E-Math tuition trains the same recognise-the-structure method these worked solutions show, taught by a team that has marked these papers for years. It runs within our weekly Secondary Math programme, Sec 1 to 4 and IP.
It is a gradient to angle (tan) question from Pythagoras & trigonometry / Mensuration, worth 10 marks: 1 + 3 + 6.
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