A man standing at point P views the top of a tower at an angle of elevation of 30°. He then walks some distance towards the tower, after which the angle of elevation of the top becomes 60°. What is the distance between the base of the tower and point P?

Introduction:
This is a height-and-distance problem involving angles of elevation. However, unlike typical questions that provide enough information to compute an exact distance, this one is designed to test whether you can identify when the given data is insufficient to determine a unique answer.

Given Data / Assumptions:

• The man initially stands at point P and sees the top of a tower at an angle of elevation of 30°.
• He walks some (unspecified) distance towards the tower and then sees the top at an angle of elevation of 60°.
• No information is given about the distance he walks or the height of the tower.
• We are asked for the distance between the base of the tower and point P.

Concept / Approach:
We typically use right-angled trigonometry for such problems. Let the height of the tower be h and the original horizontal distance from P to the base be d. After walking a distance s towards the tower, the new distance becomes d − s. The angles give us two tangent relations:
tan(30°) = h / dtan(60°) = h / (d − s)However, we have three unknowns (h, d and s) and only two equations.

Step-by-Step Solution:
Step 1: From the first position.tan(30°) = h / d ⇒ h = d * tan(30°)Step 2: From the second position.tan(60°) = h / (d − s)Step 3: Substitute h from Step 1.tan(60°) = (d * tan(30°)) / (d − s)Step 4: Observe unknowns.We now have one equation involving d and s, but there are two unknowns and no additional condition, so infinitely many (d, s) pairs satisfy the equation.

Verification / Alternative check:
If we choose any specific height h and initial distance d that satisfy tan(30°) = h / d, we can always find an s such that tan(60°) = h / (d − s). Many combinations are possible, meaning the original distance d is not uniquely determined by the given information.

Why Other Options Are Wrong:
8 units, 12 units, 10 units: Each of these gives a specific numerical distance, but there is no data to justify any of these unique values. They are arbitrary guesses.
None of these: This usually means there is some other definite numeric answer, which is not the case here. The real issue is that the data itself is insufficient.

Common Pitfalls:
Many students try to force a solution by assuming a convenient value for the height or distance without justification. Recognizing data insufficiency is an important skill and often intentionally tested in such questions.

Final Answer:
The problem does not provide enough information to determine the original distance uniquely, so the correct answer is Data inadequate.