Plane trigonometry, by S.L. Loney.

224

TRIGONOMETRY.

[Exs. XXXIV.J

27. A regular pyramid on a square base has an edge 150 feet long
and the length of the side of its base is 200 feet. Find the inclination of
its face to the base.
28. A pyramid has for base a square of side a; its vertex lies on a
line through the middle point of the base and perpendicular to it, and at
a distance h from it; prove that the angle a between the two lateral faces
is given by the equation
2h12a + 4h2
sin a=
a2 + 4h2
29. A flagstaff, 100 feet high, stands in the centre of an equilateral
triangle which is horizontal. From the top of the flagstaff each side
subtends an angle of 60~; prove that the length of the side of the triangle
is 50/6 feet.
30. The extremity of the shadow of a flagstaff, which is 6 feet high
and stands on the top of a pyramid on a square base, just reaches the
side of the base and is distant 56 and 8 feet respectively from the
extremities of that side. Find the sun's altitude if the height of the
pyramid be 34 feet.
31. The extremity of the shadow of a flagstaff, which is 6 feet high
and stands on the top of a pyramid on a square base, just reaches the
side of the base and is distant x feet and y feet respectively from the ends
of that side; prove that the height of the pyramid is
/x+y2 tan a - 6,
where a is the elevation of the sun.
32. The angle of elevation of a cloud from a point h feet above
a lake is a and the angle of depression of its reflexion in the lake is
3; prove that its height is h sin (-+ a)
sin (/3 - a)'
33. The shadow of a tower is observed to be half the known height
of the tower and sometime afterwards it is equal to the known height;
how much will the sun have gone down in the interval, given
log 2 = 30103, L tan 63~ 24' =10-3009994,

and

diff. for 1'= 3159?