Explanation
Step 1: Geometry Setup
Let OP be the vertical pillar standing perpendicular to the ground at point O.
Let h be the height of the pillar, so OP=h.
The line segment AB passes through the foot of the pillar O. This means point O lies on the straight line segment AB, separating it into two parts: AO and OB.
Given total length:
AB=AO+OB=10 meters
Step 2: Use the Angles of Elevation
The tip of the pillar P subtends angles at points A and B. Let these angles be α and β.
At point A:
α=tan−13⟹tanα=3
At point B:
β=tan−12⟹tanβ=2
Step 3: Set up Trigonometric Equations
In the right-angled triangles formed with the pillar:
In △AOP:
tanα=BasePerpendicular=AOOP
3=AOh⟹AO=3h
In △BOP:
tanβ=BasePerpendicular=OBOP
2=OBh⟹OB=2h
Step 4: Calculate the Height (h)
We know that the total length of the segment AB is the sum of AO and OB:
AO+OB=10
Substitute the expressions for AO and OB in terms of h:
3h+2h=10
Find a common denominator to combine the fractions:
62h+3h=10
65h=10
Now, isolate h:
5h=10×6
5h=60
h=560
h=12 meters
Conclusion
The height of the pillar is 12 meters.
The correct option is C) 12 meter.
