A man standing at a point P is watching the top of a tower, which makes an angle of elevation of $30^\circ$ with the man's eye. The man walks some distance towards the tower to watch its top and the angle of elevation becomes $60^\circ$. What is the distance between the base of the tower and the point P?

### Concept & Necessary Variables To solve a height and distance problem involving two angles on the same side, you need at least one linear measurement—either the height of the tower or the exact distance walked between the two points. $$ d_{\text{initial}} - d_{\text{final}} = \text{distance walked} $$ ### Step-by-Step Solution 1. Let the height of the tower be $h$. Let the initial distance from point P to the base be $x$. 2. From the initial position (Point P): $$ \tan(30^\circ) = \frac{h}{x} \Rightarrow x = h\sqrt{3} $$ 3. Let the new distance to the base after walking be $y$. 4. From the new position: $$ \tan(60^\circ) = \frac{h}{y} \Rightarrow y = \frac{h}{\sqrt{3}} $$ 5. The question asks for the distance $x$ (between the base and point P). 6. To find $x = h\sqrt{3}$, we must know $h$. However, neither the height of the tower $h$ nor the distance walked $(x - y)$ is given in the problem statement. 7. Without any single length value, the specific distance cannot be uniquely determined. ### Exam Strategy & Shortcut Scan the problem for absolute numbers. Trigonometric ratios only provide relative proportions (scales). If a question only contains angles and no actual length measurements (meters, units), finding a specific length is impossible. The answer must be "Data inadequate". ### Common Pitfall Assuming a standard height or blindly guessing a numerical answer based on the angles alone without realizing that proportional data cannot yield absolute dimensions. ### Final Answer Therefore, the correct answer is **Data inadequate**.