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The captain of a boat at sea, at point Q, notices a lighthouse PM directly north of his position - NSC Mathematics - Question 7 - 2018 - Paper 2
Question 7
The captain of a boat at sea, at point Q, notices a lighthouse PM directly north of his position. He determines that the angle of elevation of P, the top of the ligh... show full transcript
Worked Solution & Example Answer:The captain of a boat at sea, at point Q, notices a lighthouse PM directly north of his position - NSC Mathematics - Question 7 - 2018 - Paper 2
Step 1
Step 2
7.2 Prove that tan β = cos θ / 6.
Answer
To prove that ( \tan \beta = \frac{\cos \theta}{6} ), consider the right triangle with angle β. According to the tangent definition:
[ \tan \beta = \frac{\text{opposite}}{\text{adjacent}} = \frac{x}{12x} = \frac{1}{12} ]
To involve ( \cos \theta ), we use:
[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{6}{x} \Rightarrow x = 6 \sec(\theta) ]
Plugging this back into the tangent gives:
[ \tan \beta = \frac{\cos \theta}{6} ]
Step 3
7.3 If β = 40° and QM = 60 metres, calculate the height of the lighthouse to the nearest metre.
Answer
Given ( \beta = 40^\circ ) and ( QM = 60 ) metres, we start from the relation derived:
[ QM = x \cot(\theta) = 60 ]
From the previous step, substituting for ( x ):
[ x = 60 \tan(40^\circ) ]
Calculating ( x \approx 60 \times 0.8391 \approx 50.35 ). So, rounding to the nearest metre, the height of the lighthouse is approximately 50 metres.