From the observation deck of a skyscraper, Bentley measures a 48degrees 
∘
  angle of depression to a ship in the harbor below. If the observation deck is 969 feet high, what is the horizontal distance from the base of the skyscraper out to the ship? Round your answer to the nearest tenth of a foot if necessary.

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.

From the observation deck of a skyscraper, Bentley measures a 48° angle of depression to a ship in the harbor below. If the observation deck is 969 feet high, what is the horizontal distance from the base of the skyscraper out to the ship? Round your answer to the nearest tenth of a foot if necessary.

Find the horizontal distance from a 969-foot skyscraper to a ship in the harbor, given a 48° angle of depression. This problem uses trigonometric principles, specifically the tangent function, to solve real-world distance calculations. Ideal for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation