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 use trigonometric relationships in the triangles formed by the tower and the points A and B.

### Step-by-step Solution:

1. **Identify the Angles and Distances:**
- The tower subtends angles of \( 45^\circ \) at point A and \( 30^\circ \) at point B.
- The distances from the intersection point O to points A and B are given as:
- \( OA = 100 \) meters
- \( OB = 50 \) meters

2. **Set Up the Right Triangles:**
- Let \( h \) be the height of the tower PQ.
- In triangle \( PQA \) (where P is the top of the tower and A is the point on road OA):
- Using the definition of cotangent:
\[ \cot(45^\circ) = \frac{PA}{h} \]
Since \( \cot(45^\circ) = 1 \):
\[ PA = h \]

- In triangle \( PQB \) (where B is the point on road OB):
- Using the definition of cotangent:
\[ \cot(30^\circ) = \frac{PB}{h} \]
Since \( \cot(30^\circ) = \sqrt{3} \):
\[ PB = h \sqrt{3} \]

3. **Apply the Pythagorean Theorem:**
- For triangle \( OAP \):
\[ OP^2 = OA^2 + PA^2 \]
Substituting the known values:
\[ OP^2 = 100^2 + h^2 \quad \text{(1)} \]

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

4. **Set the Equations Equal:**
- Since both expressions equal \( OP^2 \), we can set them equal to each other:
\[ 100^2 + h^2 = 2500 + 3h^2 \]
- Simplifying this gives:
\[ 10000 + h^2 = 2500 + 3h^2 \]
- Rearranging terms:
\[ 10000 - 2500 = 3h^2 - h^2 \]
\[ 7500 = 2h^2 \]
\[ h^2 = \frac{7500}{2} = 3750 \]

5. **Calculate the Height \( h \):**
- 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.