Plane trigonometry, by S.L. Loney.

[Exs. VIII.]

HEIGHTS AND DISTANCES.

45

6. From the top of a cliff an observer finds that the angles of
depression of two buoys in the sea are 39~ and 26~ respectively; the
buoys are 300 yards apart and the line joining them points straight.
at the foot of the cliff; find the height of the cliff and the distance of the
nearest buoy from the foot of the cliff, given that cot 26~=2-0503, and
cot 39~= 1-2349.
7. The upper part of a tree broken over by the wind makes an angle
of 30~ with the ground, and the distance from the root to the point where
the top of the tree touches the ground is 50 feet; what was the height of
the tree?
8. The horizontal distance between two towers is 60 feet and the
angular depression of the top of the first as seen from the top of the
second which is 150 feet high is 30~; find the height of the first.
9. The angle of elevation of the top of an unfinished tower from a
point distant 120 feet from its base is 45~; how much higher must the
tower be raised so that its angle of elevation at the same point may be
60?
10. Two pillars of equal height stand on either side of a roadway
which is 100 feet wide; at a point in the roadway between the pillars the
elevations of the tops of the pillars are 60~ and 30~; find their height and
the position of the point.
11. The angle of elevation of the top of a tower is observed to be
60~; at a point 40 feet above the first point of observation the elevation
is found to be 45~; find the height of the tower and its horizontal
distance from the points of observation.
12. At the foot of a mountain the elevation of its summit is found
to be 45~; after ascending one mile up a slope of 30~ inclination the
elevation is found to be 60~. Find the height of the mountain.
13. What is the angle of elevation of the sun when the length of its
shadow is ^/3 times its height?
14. The shadow of a tower standing on a level plane is found to be
60 feet longer when the sun's altitude is 30~ than when it is 45~. Prove
that the height of the tower is 30 (1+ /3) feet.
15. On a straight coast there are three objects A, B, and C such
that AB =BC= 2 miles. A vessel approaches B in a line perpendicular
to the coast and at a certain point AC is found to subtend an angle of
600; after sailing in the same direction for ten minutes AC is found to
subtend 120~; find the rate at which the ship is going.