7. Building A is 480 feet tall and Building B is 654 feet tall. If the angle of depression from the top of Building B to the top of Building A is $$ 42^{\circ} $$, how far apart are the bulldings?

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To determine the horizontal distance between Building A and Building B, we'll use trigonometric relationships involving the angle of depression.

### Given:
- **Height of Building A (h₁):** 480 feet
- **Height of Building B (h₂):** 654 feet
- **Angle of Depression (θ):** $$42^\circ$$

### Step-by-Step Solution:

1. **Determine the Vertical Difference:**
   $$
   \Delta h = h₂ - h₁ = 654\, \text{ft} - 480\, \text{ft} = 174\, \text{ft}
   $$

2. **Visualizing the Situation:**
   - Imagine a right triangle where:
     - **Opposite Side:** Vertical difference ($$\Delta h = 174\, \text{ft}$$)
     - **Adjacent Side:** Horizontal distance between the buildings ($$x$$)
     - **Angle:** $$42^\circ$$ (angle of depression)

3. **Apply the Tangent Function:**
   The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
   $$
   \tan(\theta) = \frac{\Delta h}{x}
   $$
   Solving for $$x$$:
   $$
   x = \frac{\Delta h}{\tan(\theta)} = \frac{174}{\tan(42^\circ)}
   $$

4. **Calculate $$\tan(42^\circ)$$:**
   $$
   \tan(42^\circ) \approx 0.9004
   $$

5. **Compute the Horizontal Distance ($$x$$):**
   $$
   x \approx \frac{174}{0.9004} \approx 193\, \text{feet}
   $$

### **Conclusion:**
The horizontal distance between Building A and Building B is approximately **193 feet**.

**Answer:**  
The buildings are approximately 193 feet apart.
The buildings are approximately 193 feet apart.

7. Building A is 480 feet tall and Building B is 654 feet tall. If the angle of depression from the top of Building B to the top of Building A is \( 42^{\circ} \), how far apart are the bulldings?

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