A fire at the building B is reported by a telephone to two fire stations `F_(1)` and `F_(2)`, 10 km apart from each other on a straight road. `F_(1)` and `F_(2)` observe that the fire is at an angle of `60^(@)` and `45^(@)` respectively to the road, which station should send its team and how much will it have to travel?

Answer

To solve the problem, we will use the concept of trigonometry, specifically the tangent function, to find the distances from the fire stations to the fire.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Let the distance from fire station F1 to the fire be \( x \) km.
- Then, the distance from fire station F2 to the fire will be \( 10 - x \) km since the two stations are 10 km apart.

2. **Using the Angle from F2**:
- From fire station F2, the angle of elevation to the fire is \( 45^\circ \).
- Using the tangent function:
\[ \tan(45^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{AB}{BF2} \]
- Since \( \tan(45^\circ) = 1 \):
\[ 1 = \frac{AB}{10 - x} \]
- This implies:
\[ AB = 10 - x \quad \text{(Equation 1)} \]

3. **Using the Angle from F1**:
- From fire station F1, the angle of elevation to the fire is \( 60^\circ \).
- Again using the tangent function:
\[ \tan(60^\circ) = \frac{AB}{BF1} \]
- Since \( \tan(60^\circ) = \sqrt{3} \):
\[ \sqrt{3} = \frac{AB}{x} \]
- This implies:
\[ AB = x \sqrt{3} \quad \text{(Equation 2)} \]

4. **Equating the Two Expressions for AB**:
- From Equation 1 and Equation 2, we can set them equal to each other:
\[ 10 - x = x \sqrt{3} \]
- Rearranging gives:
\[ 10 = x \sqrt{3} + x \]
- Factoring out \( x \):
\[ 10 = x(\sqrt{3} + 1) \]
- Solving for \( x \):
\[ x = \frac{10}{\sqrt{3} + 1} \]

5. **Rationalizing the Denominator**:
- To simplify \( x \):
\[ x = \frac{10(\sqrt{3} - 1)}{(\sqrt{3} + 1)(\sqrt{3} - 1)} = \frac{10(\sqrt{3} - 1)}{3 - 1} = 5(\sqrt{3} - 1) \]
- Approximating \( \sqrt{3} \approx 1.732 \):
\[ x \approx 5(1.732 - 1) \approx 5(0.732) \approx 3.66 \text{ km} \]

6. **Finding the Distance from F2**:
- The distance from F2 to the fire is:
\[ 10 - x = 10 - 3.66 \approx 6.34 \text{ km} \]

7. **Conclusion**:
- Since \( x \approx 3.66 \) km, fire station F1 is closer to the fire.
- Therefore, **F1 should send its team**, and it will have to travel approximately **3.66 km**.