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 = 400 meters and
OB = 300 meters, than the height of the tower is

Options

A. `250sqrt2` meters
B. 500 meters
C. `50sqrt(14)` meters
D. `100sqrt7` meters

Answer

To find the height of the tower situated between the two intersecting roads OA and OB, we can use trigonometric relationships based on the angles subtended by the tower at points A and B.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Length of OA = 400 meters
- Length of OB = 300 meters
- Angle subtended at A = 45 degrees
- Angle subtended at B = 30 degrees

2. **Set Up the Problem:**
- Let the height of the tower be \( h \).
- Let \( PA \) be the horizontal distance from point A to the foot of the tower (the point directly below the tower).
- Let \( PB \) be the horizontal distance from point B to the foot of the tower.

3. **Use Trigonometric Ratios:**
- From triangle OAP (where OP is the height of the tower):
\[ \cot(45^\circ) = \frac{PA}{h} \implies 1 = \frac{PA}{h} \implies PA = h \]
- From triangle OBP:
\[ \cot(30^\circ) = \frac{PB}{h} \implies \sqrt{3} = \frac{PB}{h} \implies PB = h\sqrt{3} \]

4. **Apply the Pythagorean Theorem:**
- In triangle OAP:
\[ OP^2 = OA^2 + PA^2 \implies OP^2 = 400^2 + h^2 \]
- In triangle OBP:
\[ OP^2 = OB^2 + PB^2 \implies OP^2 = 300^2 + (h\sqrt{3})^2 \]

5. **Equate the Two Expressions for \( OP^2 \):**
\[ 400^2 + h^2 = 300^2 + 3h^2 \]

6. **Substitute the Values:**
- Calculate \( 400^2 = 160000 \) and \( 300^2 = 90000 \):
\[ 160000 + h^2 = 90000 + 3h^2 \]

7. **Rearrange the Equation:**
\[ 160000 - 90000 = 3h^2 - h^2 \implies 70000 = 2h^2 \]

8. **Solve for \( h^2 \):**
\[ h^2 = \frac{70000}{2} = 35000 \]

9. **Find \( h \):**
\[ h = \sqrt{35000} = \sqrt{350 \times 100} = 10\sqrt{350} = 10\sqrt{7 \times 50} = 10 \times 5\sqrt{14} = 50\sqrt{14} \text{ meters} \]

### Final Answer:
The height of the tower is \( 50\sqrt{14} \) meters.