plane and trigonometry
posted by rhen songalia .
a 30 foot flagstaff is fixed in the center of a circular tower 40 feet in diameter. from a point in the same horizontal plane as the foot of the tower the angles of elevation of the top of the flagstaff and the top of the tower is 30 degree repectively. find the height of the tower

You have only given one angle, but I'll call the unknown angle A and assume it is the angle to the top of the pole.
Your mission, should you choose to accept it, is to fill in the missing data and follow the logic to a final solution.
If the observer is x feet from the base of the tower (that is x+20 feet from the center of the base of the tower),
and if the height of the tower is h,
h/x = tan 30 = 1/√3
(h+30)/(x+20) = tan A
equating the two expressions for x, we get
√3 h = (h+30)/tanA  20
h = (30 cotA  20)/(√3  cotA)
If A is the angle of the tower, not the pole, then you got some fixing to do.