Plane trigonometry, by S.L. Loney.

[Exs. XXXIII.] HEIGHTS AND DISTANCES.

215

11. The elevation of a steeple at a place due south of it is 45~ and
at another place due west of it the elevation is 15~. If the distance
between the two places be a, prove that the height of the steeple is
a (^/3 -1)
2 ~/3 '
12. A person stands in the diagonal produced of the square base of
a church tower, at a distance 2a from it, and observes the angles of
elevation of each of the two outer corners of the top of the tower to be
30~, whilst that of the nearest corner is 45~. Prove that the breadth of
the tower is a (/10 - /2).
13. A person standing at a point A due south of a tower built on a
horizontal plane observes the altitude of the tower to be 60~. He then
walks to B due west of A and observes the altitude to be 45~, and again
at C in AB produced he observes it to be 30~. Prove that B is midway
between A and C.
14. At each end of a horizontal base of length 2a it is found that
the angular height of a certain peak is 0 and that at the middle point it
is q5. Prove that the vertical height of the peak is
a sin 0 sin q
^/sin (0 + 0) sin (q - 0)
15. A and B are two stations 1000 feet apart; P and Q are two
stations in the same plane as AB and on the same side of it; the angles
PAB, PBA, QAB, and QBA are respectively 75~, 30~, 45~, and 90~; find
how far P is from Q and how far each is from A and B.
For the following 4 examples a book of tables will be wanted.
16. At a point on a horizontal plane the elevation of the summit of
a mountain is found to be 22~ 15' and at another point on the plane a
mile further away in a direct line its elevation is 10~ 12'; find the height
of the mountain.
17. From the top of a hill the angles of depression of two successive
milestones, on level ground and in the same vertical plane with the
observer, are found to be 5~ and 10~ respectively. Find the height of the
hill and the horizontal distance to the nearest milestone.
18. A castle and a monument stand on the same horizontal plane.
The height of the castle is 140 feet and the angles of depression of the
top and bottom of the monument as seen from the top of the castle are
40~ and 80~ respectively. Find the height of the monument.