math

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From a lookout tower 80 ft. high, a man observes from a position 6.5 ft. below the top of the tower that the angle of elevation of the top of a certain tree is 12 degrees 40 minutes and the angle of depression of its base is 72 degrees 20 minutes. If the base of the tower and the base of the tree are at the same level, what is the height of the tree?

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If you draw a diagram, you will see that you cannot compute the height of the tree without knowing how far apart are the tower and the tree.

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The observer is 80-6.5=73.5 above the base of the tower and the base of the tree.

An angle of depression to the base of the tree of 72-20-00 means that the distance between the tower and the tree is D=73.5/tan(72-20-00)
the difference between the top of the tree and the observer is therefore
h=D sin(12-40)

The height of the tree, H, is then
H=73.5+D sin(12-40)
=73.5 + 73.5 sin(12-40-00)/tan(72-20-00)
=73.5(1+sin(12-40-00)/tan(72-20-00))
=78.6 ft

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