Plane trigonometry, by S.L. Loney.

[Exs. XXXIV.] HEIGHTS AND DISTANCES.

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7. A ship sees a lighthouse N.W. of itself. After sailing for 12 miles
in a direction 15~ south of W. the lighthouse is seen due N. Find
the distance of the lighthouse from the ship in each position.
8. A man, travelling west along a straight road, observes that when
he is due south of a certain windmill the straight line drawn to a distant
church makes an angle of 30~ with the road. A mile further on the
bearings of the windmill and tower are respectively N.E. and N.W. Find
the distances of the tower from the windmill and from the nearest point
of the road.
9. An observer on a headland sees a ship due north of him; after a
quarter of an hour he sees it due east and after another half-hour he sees
it due south-east; find the direction that the ship's course makes with
the meridian and the time after the ship is first seen until it is nearest
the observer, supposing that it sails uniformly in a straight line.
10. A man walking along a straight road which runs in a direction
30~ east of north notes when he is due south of a certain house; when he
has walked a mile further he observes that the house lies due west and
that a windmill on the opposite side of the road is N.E. of him; three
miles further on he finds that he is due north of the windmill; prove
that the line joining the house and the windmill makes with the road
the angle whose tangent is
48 - 25,3
11 '
11. A, B, and C are three consecutive milestones on a straight road
from each of which a distant spire is visible. The spire is observed to
bear north-east at A, east at B, and 60~ east of south at C. Prove that
the shortest distance of the spire from the road is 7 + miles.
13
12. Two stations due south of a tower, which leans towards the
north, are at distances a and b from its foot; if a and (3 be the
elevations of the top of the tower from these stations, prove that its
inclination to the vertical is
b cot a - a cot
cot - l
b-a
13. From a point A on a level plane the angle of elevation of a
balloon is a, the balloon being south of A; from a point B which is at a
distance C south of A the balloon is seen northwards at an elevation of.; find the distance of the balloon from A and its height above the
ground.