To solve this problem, we can use trigonometry, specifically the tangent function, since we are given angles of depression and the height of the observer above sea level.

Step-by-Step Approach:

1. Understanding the situation:

   • The observer is 350 feet above sea level.
   • There are two ships offshore, and the angles of depression from the observer to the two ships are $\beta = 5^\circ$ and $\theta = 9.5^\circ$.
   • We need to find how far apart the two ships are horizontally.

2. Using Trigonometry: The angles of depression are measured from the horizontal line down to the ships. So, for each ship, we can use the tangent of the angle of depression to relate the height of the lighthouse and the horizontal distance to the ships.

   For the first ship (with angle $\theta = 9.5^\circ$):

   $\tan(\theta) = \frac{\text{height of lighthouse}}{\text{distance to first ship}}$

   $\tan(9.5^\circ) = \frac{350}{d_1}$

   Solving for $d_1$:

   $d_1 = \frac{350}{\tan(9.5^\circ)}$

   Similarly, for the second ship (with angle $\beta = 5^\circ$):

   $\tan(\beta) = \frac{350}{d_2}$

   Solving for $d_2$:

   $d_2 = \frac{350}{\tan(5^\circ)}$

3. Finding the horizontal distance between the two ships: The distance between the two ships will be the difference between $d_1$ and $d_2$, i.e.

   $\text{Distance between ships} = d_2 - d_1$

Let me calculate this.The horizontal distance between the two ships is approximately 1909.0 feet.

Would you like further details or explanations on any part of the solution?

Here are 5 related questions you might find useful:

1. How would the problem change if the lighthouse height were different?
2. What if the angles of depression were larger, how would that affect the distance?
3. Can we solve this problem using the sine function instead of tangent?
4. How do we calculate the angle of depression if we are given the distance between ships?
5. How does changing the angle unit from degrees to radians affect the outcome?

Tip: Always ensure your calculator is set to the correct mode (degrees/radians) when solving trigonometric problems.