The angular elevation of a tower from a point is $\displaystyle 30^{\circ}$ As we more $100$ m nearer to the base of the tower the angle of elevation becomes $\displaystyle 60^{\circ}$. Find the height of the tower and the distance of the first point from the base of the tower

Let h be the height of the tower and d be the initial distance from the base. Using the 30 degree angle of elevation, tan(30) = h divided by d, so d equals h times sqrt(3). After moving 100 m closer, the new distance is d minus 100, and tan(60) = h divided by (d minus 100). Substituting d gives sqrt(3) = h divided by (h times sqrt(3) minus 100). Solving this equation yields a height h of 150 m, which corresponds to the first value requested.

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