Theory question 109
Trigonometry(a) The angle of elevation of the top of a 75 m tower from a ship and a boat are 26.4° and 57.4° respectively. Given that, the ship, boat and the foot of the tower are on the same horizontal: (i) Illustrate the information in a diagram (ii) Calculate, correct to three significant figures, the distance between the ship and the boat.
(b) A sector of a circle has a radius of 18 cm and subtends an angle of 120° at the centre of the circle. Find, correct to five significant figures, the area of the sector. [Take π = 22/7]
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(a)(i) Vertical tower 75 m; ship and boat on the ground line, ship further away (26.4°), boat nearer (57.4°)
(ii) Ship to tower = 75 ÷ tan 26.4° ≈ 151.09 m; boat to tower = 75 ÷ tan 57.4° ≈ 47.96 m
Same side of the tower: distance ≈ 151.09 − 47.96 ≈ 103 m (if on opposite sides: 151.09 + 47.96 ≈ 199 m)
(b) Area = 120/360 × 22/7 × 18² = 339.43 cm²
The question does not say which side; the usual reading is that both are on the same side of the tower.