Plane trigonometry with practical applications, by Leonard E. Dickson.

10

TRIGONOMETRY

[Art. 8

EXAMPLE 3. The angle of elevation of a balloon B from a station P due
south of it is 60~, and from another station Q due west from P and 2 miles
from it the angle of elevation is 45~. Find the
height h of the balloon.
Solution. Let F be the foot of the perpendicular
from B to the ground (Fig. 13). In triangle BFP,
/ h \ Z F is a right angle and
3 = tan 60~ = -
FP
/whence FP = hll/3. In triangle BFQ, Z F is a
right angle, and hence ZB = 45~, so that QF = h.
H i~ Finally, in triangle FPQ, ZP is a right angle, so
Q l- that
___ h2
FIG. 13 Q2 = FP2 + 4, h2 = -h + 4, h2 = 6, h = 6.
EXERCISES ON HEIGHTS AND DISTANCES
(Give exact answers, not computing square roots.)
1. At the top of a house 80 feet high, the angle of elevation of the
top of a tower is 45~; on the ground floor it is 60~. Find the height of
the tower.
2. A man 6 feet tall observes that the angle of elevation of the top of
a tree is 45~, while that of the point where the branches begin is 30~.
The latter point is 17 feet above the ground. How high is the tree?
3. The angle of ascent of a road is 30~. If a man travels 500 feet up
the road, how many feet has he risen?
4. The shadow of a flagpole standing on level ground is 50 feet
longer when the angle of elevation of the sun is 30~ than when it is 45~.
Find the height of the pole.
5. The upper part of a tree broken over by the wind makes an angle
of 30~ with the ground and its top rests on the ground at a point 75 feet
from the root. What was the height of the tree?
6. From a boat at sea the angles of elevation of the top and base of a
tree 100 feet high on the top of a bluff are found to be 60~ and 45~ respectively. Find the height of the bluff.
7. At a point half way between two buildings the angles of elevation of their tops are 30~ and 60~. Prove that one building is three times
as high as the other.