Theory question 110
Bearings(a) A cottage is on a bearing of 200° and 110° from Dogbe's and Manu's farms respectively. If Dogbe walked 5 km and Manu 3 km from the cottage to their farms, find, correct to: (i) two significant figures, the distance between the two farms; (ii) the nearest degree, the bearing of Manu's farm from Dogbe's.
(b) A ladder 10 m long leaned against a vertical wall x m high. The distance between the wall and the foot of the ladder is 2 m longer than the height of the wall. Calculate the value of x.
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(a) From the cottage, Dogbe's farm is on 020° (5 km) and Manu's on 290° (3 km); the angle between them = 290° − 20° = 270°, i.e. 90°
(i) Distance = √(52 + 32) = √34 ≈ 5.8 km
(ii) At Dogbe's farm, tan θ = 3/5 → θ ≈ 31.0°; bearing of Manu's farm = 200° + 31° ≈ 231°
(b) x2 + (x + 2)2 = 102 → 2x2 + 4x − 96 = 0 → x2 + 2x − 48 = 0
(x + 8)(x − 6) = 0 → x = 6 (length must be positive)
Turn each bearing round by 180° to get the direction from the cottage to each farm.