To find the horizontal distance from the base of the skyscraper to the ship, we can use trigonometry. Specifically, we’ll use the tangent function, which relates an angle in a right triangle to the ratio of the opposite side (height of the skyscraper) to the adjacent side (horizontal distance to the ship).

Step-by-Step Solution

1. Identify Given Information:

   • Angle of depression from the observation deck to the ship: $48^\circ$
   • Height of the skyscraper (opposite side in the triangle): $969$ feet

2. Set Up the Tangent Equation:

   • Let $d$ represent the horizontal distance from the base of the skyscraper to the ship.
   • We know that: $\tan(48^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{969}{d}$

3. Solve for $d$:

   • Rearrange the equation to isolate $d$: $d = \frac{969}{\tan(48^\circ)}$

4. Calculate $d$:

   • Using a calculator to find $\tan(48^\circ) \approx 1.1106$, $d = \frac{969}{1.1106} \approx 872.5 \text{ feet}$

Final Answer

The horizontal distance from the base of the skyscraper to the ship is approximately 872.5 feet.

Would you like a more detailed breakdown or have any questions about the process?

Here are some related questions that could further expand this topic:

1. What would be the distance if the angle of depression was 60 degrees?
2. How would the calculation change if the height of the skyscraper were doubled?
3. Can this method be used to calculate distances if the angle is very small, say 5 degrees?
4. What if the observation deck height was measured in meters instead of feet?
5. How accurate are these trigonometric calculations in real-life surveying?

Tip: For problems involving angles of elevation or depression, drawing a diagram can often clarify the relationships between the angles and sides.