A man is observing a yacht from the top of a cliff. The angle of depression of the yatch from the top of the cliff is 44. If the yatch sails a further 200 meters further away from the cliff, the angle of depression is now 38. How high is the cliff?
+130514 We can use the tangent function here......let h be the height of the cliff and d be the initial distance from the boat to the base of the cliff
And we have, in the intial observaation:
tan 44 = h / d → h = d tan 44
And, in the second observation, we have
tan 38 = h / [ d + 200 ] = [ d tan 44] / [ d + 200] multiply both sides by [ d + 200]
[d + 200] tan 38 = d tan 44
d tan 38 + 200 tan 38 = d tan 44 rearrange
d tan 38 - d tan 44 = - 200 tan 38 factor out d on the left
d [ tan 38 - tan 44] = - 200 tan 38 divide both sides by [ tan 38 - tan 44]
d = -200 tan38 / [ tan 38 - tan 44] = about 847.4 m
And the heght of the cliff = (847.4) tan (44) = about 818.3 m
