Two vertical poles are 150 m apart and the height of one is three times that of the other if from the middle point of the line joining their feet , an observer finds the angles of elvation of their tops ot be complememntary then the height of the shorter pole ( in meters ) is :

Options

A. ` 20 sqrt(3)`
B. ` 25 sqrt(3)`
C. `30`
D. `25`

Answer

To solve the problem, we will follow these steps:

### Step 1: Define Variables
Let the height of the shorter pole be \( x \) meters. Therefore, the height of the taller pole will be \( 3x \) meters.

### Step 2: Setup the Geometry
The two poles are 150 meters apart. The observer is standing at the midpoint between the two poles, which means he is 75 meters away from each pole.

### Step 3: Angles of Elevation
Let the angle of elevation to the top of the shorter pole be \( \theta \) and the angle of elevation to the top of the taller pole be \( 90^\circ - \theta \) (since they are complementary).

### Step 4: Write the Tangent Equations
For the shorter pole:
\[ \tan(\theta) = \frac{x}{75} \]
For the taller pole:
\[ \tan(90^\circ - \theta) = \cot(\theta) = \frac{3x}{75} \]

### Step 5: Express Cotangent in Terms of Tangent
From the first equation, we can express \( \cot(\theta) \):
\[ \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{75}{x} \]

### Step 6: Set Up the Equation
Now we equate the two expressions for \( \cot(\theta) \):
\[ \frac{3x}{75} = \frac{75}{x} \]

### Step 7: Cross Multiply
Cross multiplying gives:
\[ 3x^2 = 75 \times 75 \]

### Step 8: Solve for \( x^2 \)
\[ 3x^2 = 5625 \]
\[ x^2 = \frac{5625}{3} = 1875 \]

### Step 9: Find \( x \)
Taking the square root:
\[ x = \sqrt{1875} = 25\sqrt{3} \]

### Step 10: Conclusion
Thus, the height of the shorter pole is:
\[ \text{Height of the shorter pole} = 25\sqrt{3} \text{ meters} \]