In aptitude (height and distance simplification), the angle of elevation of the top of a tower from point A is 30 degrees. After moving 20 meters towards the foot of the tower to point B, the angle of elevation becomes 60 degrees. Using basic trigonometry, find the height of the tower in meters.

Introduction / Context:
This is a standard height and distance problem in trigonometry, frequently tested in aptitude and engineering entrance examinations. The key idea is to form right-angled triangles from the given angles of elevation and to express the height of the tower and horizontal distances in terms of tangent functions. By using two different positions with known separation, we can set up two equations and solve for the unknown height.

Given Data / Assumptions:
- The angle of elevation from point A to the top of the tower is 30 degrees.
- After moving 20 meters towards the tower to point B, the angle of elevation becomes 60 degrees.
- The tower stands vertically on level ground, and the lines of sight form right angles with the horizontal ground at the base of the tower.
- We need to find the height h of the tower.

Concept / Approach:
We use the tangent function in right-angled triangles. If h is the height of the tower and x is the horizontal distance from point A to the foot of the tower, then from point A we have tan(30 degrees) = h / x. From point B, the new distance to the tower is (x - 20), and tan(60 degrees) = h / (x - 20). These two equations in h and x can be solved simultaneously to find the height h.

Step-by-Step Solution:
Step 1: Let h be the height of the tower and x be the distance from point A to the foot of the tower.Step 2: From point A, tan(30 degrees) = h / x. We know tan(30 degrees) = 1 / sqrt(3), so h / x = 1 / sqrt(3), giving h = x / sqrt(3).Step 3: From point B, which is 20 meters closer, the distance to the tower is x - 20. The angle of elevation is 60 degrees.Step 4: So tan(60 degrees) = h / (x - 20). We know tan(60 degrees) = sqrt(3), so h / (x - 20) = sqrt(3), giving h = sqrt(3) * (x - 20).Step 5: Equate the two expressions for h: x / sqrt(3) = sqrt(3) * (x - 20).Step 6: Multiply both sides by sqrt(3): x = 3 * (x - 20).Step 7: Expand the right-hand side: x = 3x - 60.Step 8: Rearrange to solve for x: 3x - x = 60, so 2x = 60, hence x = 30.Step 9: Substitute x back into h = x / sqrt(3): h = 30 / sqrt(3) = 10 * sqrt(3).

Verification / Alternative check:
Check from point B: distance to tower is x - 20 = 30 - 20 = 10 meters. For height h = 10 * sqrt(3), tan(60 degrees) should be h / 10 = (10 * sqrt(3)) / 10 = sqrt(3), which matches tan(60 degrees). From point A: h / x = (10 * sqrt(3)) / 30 = sqrt(3) / 3 = 1 / sqrt(3), which matches tan(30 degrees). Both checks confirm the correctness of the found height.

Why Other Options Are Wrong:
- sqrt(3) and 5*sqrt(3): These values are far too small to be consistent with the geometry of the problem, especially considering the 20 meter horizontal shift.
- 20*sqrt(3) and 15*sqrt(3): These are larger than required and do not satisfy both tangent equations simultaneously. Substituting them into tan(30 degrees) and tan(60 degrees) relationships would fail either at point A or point B.

Common Pitfalls:
Common mistakes include mixing up which distance is x and which is x - 20, writing tan(30 degrees) and tan(60 degrees) incorrectly, or forgetting that tan(30 degrees) = 1 / sqrt(3) and tan(60 degrees) = sqrt(3). Some students also assume that the height is numerically equal to one of the given distances without solving the equations properly. Always write the equations explicitly and solve systematically.

Final Answer:
The height of the tower is 10*sqrt(3) meters.