Plane trigonometry, by S.L. Loney.

[Exs. XXXIV.] HEIGHTS AND DISTANCES.

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34. An isosceles triangle of wood is placed in a vertical plane, vertex
upwards, and faces the sun. If 2a be the base of the triangle, h its
height, and 30~ the altitude of the sun, prove that the tangent of the angle
at the apex of the shadow is 3h.-a-3
3h2 -a2
35. A rectangular target faces due south, being vertical and standing
on a horizontal plane. Compare the area of the target with that of its
shadow on the ground when the sun is 3~ from the south at an altitude
of a~.
36. A spherical ball, of diameter 8, subtends an angle a at a man's
eye when the elevation of its centre is 3; prove that the height of the
1.a
centre of the ball is 2 8 sin p cosec.
2 2
37. A man standing a plane observes a row of equal and equidistant pillars, the 10th and 17th of which subtend the same angle that
they would do if they were in the position of the first and were
1 1
respectively - and of their height. Prove that, neglecting the height
2 3
of the man's eye, the line of pillars is inclined to the line drawn to the
first at an angle whose secant is nearly 2'6.
For the following 4 examples a book of tables will be wanted.
38. A and B are two points on the opposite bank of a river 1000 feet
wide and between them is the mast of a ship PN; the vertical elevation
of P at A is 14~ 20' and at B it is 8~ 10'. What is the height of P
above AB?
39. AB is a line 1000 yards long; B is due north of A and from B
a distant point P bears 70~ east of north; at A it bears 41~ 22' east of
north; find the distance from A to P.
40. A is a station exactly 10 miles west of B. The bearing of a
particular rock from A is 74~ 19' east of north and its bearing from B is
26~ 51' west of north. How far is it north of the line AB?
41. The summit of a spire is vertically over the middle point of a
horizontal square enclosure whose side is of length a feet; the height of
the spire is h feet above the level of the square. If the shadow of the
spire just reach a corner of the square when the sun has an altitude 0,
prove that
h7,/2=a tan 0.
Calculate h, having given a= 1000 feet and 0 = 25~15'.

L. T.

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