The given diagram shows two boats (B and A) and a lighthouse (L). The angles of elevation from the boats to the top of the lighthouse are $2^\circ$ (from B) and $12^\circ$ (from A). Additionally, the horizontal distance from A to L is given as 804 meters. We are tasked with finding the distance $BA$.

To solve this problem, we will use trigonometric relationships in right triangles:

1. Label the heights and distances:

   • The height of the lighthouse is $h$.
   • The horizontal distance from B to L is $x + 804$, where $x$ is the distance between B and A.

2. Using the tangent function:

   • From boat A: $\tan(12^\circ) = \frac{h}{804}.$ So, $h = 804 \cdot \tan(12^\circ).$

   • From boat B: $\tan(2^\circ) = \frac{h}{x + 804}.$ Substituting $h = 804 \cdot \tan(12^\circ)$ into this equation: $\tan(2^\circ) = \frac{804 \cdot \tan(12^\circ)}{x + 804}.$

3. Solving for $x$: Rearranging: $x + 804 = \frac{804 \cdot \tan(12^\circ)}{\tan(2^\circ)}.$ Therefore: $x = \frac{804 \cdot \tan(12^\circ)}{\tan(2^\circ)} - 804.$

4. Calculate the value of $x$: Let's compute this value. math

Given values

angle_A = 12 # degrees angle_B = 2 # degrees distance_AL = 804 # meters

Calculations

h = distance_AL * math.tan(math.radians(angle_A)) # Height of the lighthouse x = (distance_AL * math.tan(math.radians(angle_A)) / math.tan(math.radians(angle_B))) - distance_AL # Distance BA

h, xThe height of the lighthouse ($h$) is approximately 170.90 meters, and the distance between the two boats ($BA$) is approximately 4089.81 meters.

Let me know if you would like further explanation or calculations! Here are related questions you might find useful:

1. What is the height of the lighthouse if the angles were changed to $3^\circ$ and $15^\circ$?
2. How would the calculation change if the distance $AL$ was different?
3. How can trigonometry be used to determine angles in navigation?
4. What other methods besides trigonometry could solve this problem?
5. How does the accuracy of the angles affect the final result?

Tip: Always ensure your calculator is set to the correct mode (degrees or radians) when performing trigonometric calculations!