On level ground, the angle of elevation of the top of a tower is 30°. After moving 20 m nearer the tower, the angle increases to 60°. Find the height of the tower (in metres).
Aptitude
Height and Distance
Difficulty: Medium
Choose an option
-
A10 m
-
B10√3 m
-
C15 m
-
D20 m
-
E30 m
Answer
Correct Answer: 10√3 m
Explanation
Introduction / Context:Two elevation angles from positions 20 m apart allow solving the system for both distance and height via tangent relations.
Given Data / Assumptions:
- Initial angle = 30° at distance x.
- Second angle = 60° at distance x − 20.
- Height h unknown.
Concept / Approach:Write h = x * tan 30° and h = (x − 20) * tan 60°. Equate to eliminate h and solve for x, then back-substitute to get h.
Step-by-Step Solution:
x * (1/√3) = (x − 20) * √3.x = 3(x − 20) ⇒ x = 3x − 60 ⇒ 2x = 60 ⇒ x = 30 m.h = x / √3 = 30/√3 = 10√3 m.Verification / Alternative check:At x − 20 = 10 m, tan 60° = h/10 ⇒ h = 10√3 ✔; all consistent.
Why Other Options Are Wrong:10, 15, 20, 30 m ignore √3 relationships intrinsic to 30°/60° pairs.
Common Pitfalls:Using x + 20 instead of x − 20 for “moved nearer”; mixing degrees and radians in calculators.
Final Answer:10√3 m