a point on one bank he finds the angle of elevation of the top of a tree on the other bank to be 60 degress. from his initial observation point he walks back 200 feet in a straight line with the tree and his first observation point and finds the angle of elevation to the top of the tree to be 40 degress. what is the width of the river?
We need to form 2 rt triangles with a common ver. side.
First, draw a rectangle with the long
side parallel to the x-axis. Then draw
a diagonal from the lower Lt corner to
the upper rt corner. This is the hyp
of the larger triangle. From a point near the center of the hor side, draw
another hyp to the upper rt corner. This is the hyp of the smaller
triangle.
The angle bet. the hyp of larger tri-
angle and hor is 40 deg. The 60-degree
angle is bet hyp and hor of smaller triangle.
The dist. bet. the 2 hyp is 200 ft Label the remaining dist. X ft and the
ver side is h for ht of tree.
Width = 200 + X.
tan60 = h/x,
h = xtan60.
tan40 = h/(200+x),
h = (x+200)tan40.
xtan60 = (x+200)tan40,
xtan60 = xtan40 + 200tan40,
xtan60 - xtan40 = 200tan40,
1.732x - 0.8391x = 167.82,
0.8929x = 167.82,
X = 188.
Width = 200 + 188 = 388 Ft.