Question

A student measures the angle of elevation of a cloud to be 65° at point A. Then the student moves 100 feet in the direction of the cloud and takes a second observation at point B. At point B the elevation is 75. Using this information find the height, to the nearest foot, of the cloud directly above the ground.

Answer 1

see the attached figure to better understand the problem

we know that
in the triangle ACD

tan 65°=CD/(100+x)------> CD=(100+x)*tan 65°------> equation 1

in the triangle BCD

tan 75°=CD/x---------> CD=x*tan 75°------> equation 2

equals 1 and 2
(100+x)*tan 65°=x*tan 75°-----> (100+x)*2.14=x*3.73
100*tan 65+x*tan 65=x*tan 75-----> x*{tan 75-tan 65]=100* tan 65
x=100* tan 65/[tan 75- tan 65]------> x=135.08 ft

CD=x*tan 75°------->CD=135.08*tan 75°------> CD=504.12 ft

the answer is
the height of the cloud directly above the ground is 504 ft