On a sunny day, a tower casts its shadow across a field. At 10:00 AM, the shadow of the tower reaches point P. By 4:00 PM, the shadow stretches to point Q. and the distance between P and Q 14 measures 80 meters. During this time, the angle of the sun's elevation changes from 60 degrees to 30 degrees, then determine the height of the tower?

Let the height of the tower be h and the distance from its base to point Q be x, so tan 30 degrees equals h divided by (x + 80) and tan 60 degrees equals h divided by x. This gives x equals h divided by root 3 and x + 80 equals root 3 times h. Substituting the first into the second yields h divided by root 3 + 80 equals root 3 times h, which simplifies to 80 equals 2h divided by root 3, making the height 40 divided by root 3, or 20 root 3 meters.

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