Plane trigonometry, by S.L. Loney.

222 TRIGONOMETRY. [EXS. XXXIV.]
14. A statue on the top of a pillar subtends the same angle a at
distances of 9 and 11 yards from the pillar; if tan a =~, find the height
of the pillar and of the statue.
15. A tower and a spire on the top of the tower subtend equal angles
at a point whose distance from the foot of the tower is a; if h be the
height of the tower, prove that the height of the spire is
a2 + h2
ha2 -h2
16. A flagstaff on the top of a tower is observed to subtend the same
angle at two points on a horizontal plane, which lie on a line passing
through the centre of the base of the tower and whose distance from one
another is 2a, and an angle f at a point halfway between them. Prove
that the height of the flagstaff is
a sin a 2 sin a
aa / 2sin a
cos a sin (a-/p)
17. An observer in the first place stations himself at a distance a
feet from a column standing upon a mound. He finds that the column
subtends an angle, whose tangent is 2, at his eye which may be supposed
to be on the horizontal plane through the base of the mound. On
2
moving -a feet nearer the column he finds that the angle subtended is
unchanged. Find the height of the mound and of the column.
18. A church tower stands on the bank of a river which is 150 feet
wide and on the top of the tower is a spire 30 feet high. To an observer
on the opposite bank of the river the spire subtends the same angle that
a pole six feet high subtends when placed upright on the ground at the
foot of the tower. Prove that the height of the tower is nearly 285 feet.
19. A person, wishing to ascertain the height of a tower, stations
himself on a horizontal plane through its foot at a point at which the
elevation of the top is 30~. On walking a distance a in a certain direction
he finds that the elevation of the top is the same as before, and on then
walking a distance 5 a at right angles to his former direction he finds the
elevation of the top to be 60~. Prove that the height of the tower is
either /6a or a.
'V6 ' - 48