Step 1: Define Variables
Let the height of the pillar be \( h \).
Let the foot of the pillar be \( O \).
Let \( A \) and \( B \) be two points on the ground such that \( AB = 10 \) meters.
Let \( OA = x \) and \( OB = 10 - x \).
From the given angles: \[ \tan(\theta_A) = 3 \quad \Rightarrow \quad \frac{h}{x} = 3 \quad \Rightarrow \quad h = 3x \] \[ \tan(\theta_B) = 2 \quad \Rightarrow \quad \frac{h}{10-x} = 2 \quad \Rightarrow \quad h = 2(10 - x) \] Step 2: Solve for \( h \)
Equating the two expressions for \( h \): \[ 3x = 2(10 - x) \] \[ 3x = 20 - 2x \] \[ 3x + 2x = 20 \] \[ 5x = 20 \] \[ x = 4 \] Substituting \( x = 4 \) into \( h = 3x \): \[ h = 3(4) = 12 \] Step 3: Verify the Answer
Substituting \( h = 12 \) into the second equation: \[ 12 = 2(10 - 4) \] \[ 12 = 12 \] The equation holds, confirming that the height of the pillar is 12 meters.