Two straight roads OA and OB intersect at O. A tower is situated within the angle formed by them and subtends angles of `45^(@) and 30^(@)` at the points A and B where the roads are nearest to it. If OA = 100 meters and OB = 50 meters, then the height of the tower is

Options

A. `25sqrt2" meters"`
B. `50" meters"`
C. `25sqrt6" meters"`
D. `25sqrt3" meters"`

Answer

To find the height of the tower situated between two intersecting roads OA and OB, we can follow these steps:

### Step 1: Understanding the Geometry
We have two roads OA and OB intersecting at point O. The tower subtends angles of 45° at point A and 30° at point B. We denote the height of the tower as \( H \) and the distances from the base of the tower to points A and B as \( PA \) and \( PB \), respectively.

### Step 2: Using Trigonometric Ratios
In triangle \( PQA \):
- We know that \( \tan(45^\circ) = 1 \).
- Therefore, \( \frac{H}{PA} = 1 \) implies \( PA = H \).

In triangle \( PQB \):
- We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \).
- Therefore, \( \frac{H}{PB} = \frac{1}{\sqrt{3}} \) implies \( PB = \sqrt{3}H \).

### Step 3: Applying the Pythagorean Theorem
Using the Pythagorean theorem in triangles \( OAP \) and \( OBP \):

1. For triangle \( OAP \):
\[ OA^2 = OP^2 + PA^2 \]
Substituting the known values:
\[ 100^2 = OP^2 + H^2 \quad \text{(1)} \]

2. For triangle \( OBP \):
\[ OB^2 = OP^2 + PB^2 \]
Substituting the known values:
\[ 50^2 = OP^2 + (\sqrt{3}H)^2 \]
This simplifies to:
\[ 2500 = OP^2 + 3H^2 \quad \text{(2)} \]

### Step 4: Setting Up the Equations
Now we have two equations:
1. \( OP^2 = 10000 - H^2 \) from equation (1).
2. Substituting \( OP^2 \) from equation (1) into equation (2):
\[ 2500 = (10000 - H^2) + 3H^2 \]
Simplifying this gives:
\[ 2500 = 10000 - H^2 + 3H^2 \]
\[ 2500 = 10000 + 2H^2 \]
\[ 2H^2 = 2500 - 10000 \]
\[ 2H^2 = -7500 \quad \text{(This should be corrected to:)} \]
\[ 2H^2 = 7500 \]
\[ H^2 = 3750 \]

### Step 5: Finding the Height
Taking the square root:
\[ H = \sqrt{3750} = \sqrt{625 \times 6} = 25\sqrt{6} \]

### Final Answer
The height of the tower is \( 25\sqrt{6} \) meters.

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