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To solve this problem, we can use trigonometry. The angle of depression forms a right triangle, with the height of the skyscraper as the vertical leg and the horizontal distance to the ship as the horizontal leg. The angle of depression (48°) is given, and we know the height of the skyscraper is 871 feet.

To find the horizontal distance \( x \), we can use the tangent function, which relates the angle of a right triangle to the ratio of the opposite side (the height of the skyscraper) to the adjacent side (the horizontal distance):

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]

Here:
- \( \theta = 48^\circ \)
- Opposite side = 871 feet (height of the skyscraper)
- Adjacent side = \( x \) (the horizontal distance to the ship)

So,

\[
\tan(48^\circ) = \frac{871}{x}
\]

Rearranging the equation to solve for \( x \):

\[
x = \frac{871}{\tan(48^\circ)}
\]

Now, let me calculate that value for you.The horizontal distance from the base of the skyscraper to the ship is approximately 784.25 feet.

Do you want further details or have any questions?

Here are some related questions:
1. How would the answer change if the angle of depression was smaller, say 30°?
2. What if the skyscraper height was doubled to 1,742 feet but the angle remained 48°?
3. How can we use the sine function to solve this problem?
4. Could this problem be solved using the cosine rule? Why or why not?
5. What are other practical applications of using angles of depression in real-world scenarios?

**Tip:** In right-triangle problems, always consider the trigonometric function that directly relates the known and unknown sides of the triangle.

From the observation deck of a skyscraper, Ian measures a 48° angle of depression to a ship in the harbor below. If the observation deck is 871 feet high, what is the horizontal distance from the base of the skyscraper out to the ship? Round your answer to the nearest hundredth of a foot if necessary.

Learn how to find the horizontal distance from the base of a skyscraper to a ship in the harbor using trigonometry. This problem involves an angle of depression of 48°, a skyscraper height of 871 feet, and uses the tangent function to calculate the horizontal distance. Suitable for students in Grades 9-12.

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