The angles of elevation of the top of a tower observed from two points on the ground are $30^o$ and $45^o$. If the distance between the two points is $20$metres, then the height of the tower is

Let the total distance from the 45-degree observation point to the tower be x, making the distance from the 30-degree observation point x + 20. Using the tangent ratios, tan(45 degrees) = h/x giving h = x, and tan(30 degrees) = h/(x + 20) giving 1/sqrt(3) = x/(x + 20). Solving x + 20 = x*sqrt(3) yields x = 20/(sqrt(3) - 1), which rationalizes to 10*(sqrt(3) + 1), or approximately 27.3 m for the height.

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