A man is standing at a point P and observes the top of a vertical tower. The angle of elevation from point P to the top of the tower is 30°. He then walks some distance straight towards the tower, and now the angle of elevation to the top becomes 60°. What is the distance between the base of the tower and the point P?

Introduction / Context:
This question deals with height and distance using trigonometry, where a man observes a tower from two different positions. The angles of elevation change, but the actual distances walked are not specified. Such questions are meant to test not only trigonometric skills but also your understanding of when information is sufficient or insufficient to determine a unique numerical answer. Recognizing data inadequacy is as important as performing calculations correctly.

Given Data / Assumptions:

• The tower is vertical and stands on level ground.
• From point P, the angle of elevation to the top of the tower is 30°.
• The man walks some distance directly towards the tower.
• From the new point, the angle of elevation to the top becomes 60°.
• No numeric value of the walking distance or the height of the tower is given.
• We are asked for the distance between the base of the tower and the original point P.

Concept / Approach:
If we let x be the initial distance from P to the tower base and h be the height of the tower, then using basic trigonometry: tan(30°) = h / x and tan(60°) = h / (x - d), where d is the distance walked towards the tower. These equations give relationships between h, x, and d. However, unless we are given an actual numeric value for h or d, or a further condition, we will not be able to solve for x uniquely. The system of equations only provides ratios, not absolute measures.

Step-by-Step Reasoning:
Step 1: Set up variables. Let the height of the tower be h. Let the initial distance from P to the tower be x. Let the man walk a distance d towards the tower, so the new distance is x - d. Step 2: Use the first angle of elevation. tan(30°) = h / x. Since tan(30°) = 1 / √3, we get h = x / √3. Step 3: Use the second angle of elevation. tan(60°) = h / (x - d). tan(60°) = √3, so h = √3 (x - d). Step 4: Equate the two expressions for h. x / √3 = √3 (x - d). Multiply both sides by √3: x = 3(x - d) = 3x - 3d. Step 5: Rearrange to relate x and d. x = 3x - 3d implies 2x = 3d, so d = (2/3)x. We now only know that d is two thirds of x, but we do not know their actual numeric values.

Verification / Alternative check:
The equations derived show a proportional relationship between x and d but do not fix their magnitudes. For example, if x were 3 units, then d would be 2 units; if x were 30 units, then d would be 20 units. In both cases, the angles of 30° and 60° could be satisfied with different tower heights. Thus, there is an infinite family of solutions that satisfy the trigonometric conditions, and no unique numeric value of x can be determined from the given information alone.

Why Other Options Are Wrong:
8 units and 12 units: These are specific distances that cannot be justified because the equations do not yield a single numeric value for x. Without additional constraints, choosing any particular distance is arbitrary.
None of these: This is less precise because we actually have a clear reason that the data is insufficient; the correct description is that the data is inadequate or that we cannot determine a unique numeric answer.
Cannot be determined exactly: This is essentially similar in meaning to data inadequate, but the standard exam phrasing here is best captured by the option explicitly stating data inadequacy.

Common Pitfalls:
Many students try to assume a specific value for the distance walked or the height of the tower, even though the question does not provide such information. This leads to arbitrary calculations and incorrect answers. Another pitfall is to assume that two equations always guarantee a unique solution, without checking how many unknowns are involved. In this case, there are three unknowns (h, x, d) but only two independent equations, so one degree of freedom remains, preventing a unique answer for x.

Final Answer:
The information given is not sufficient; the distance from the base of the tower to point P is data inadequate.