length of bridge is 5035 ft. angles of elevation 76 degrees and 74 degrees. how deep is canyon

To find the depth of the canyon using the information provided, we can use trigonometric relationships based on the angles of elevation from both ends of the bridge.

### Step-by-Step Solution:

1. **Set Up the Problem:**
   - Let \( A \) and \( B \) be the two points on the ground directly below the ends of the bridge.
   - Let \( C \) be the point where the bridge meets the canyon (the top).
   - The angles of elevation from points \( A \) and \( B \) to point \( C \) are \( 76^\circ \) and \( 74^\circ \), respectively.
   - The length of the bridge \( AB = 5035 \) ft.

2. **Form Right Triangles:**
   - From point \( A \) (angle \( 76^\circ \)), we have a right triangle where:
     \[
     h = AC = AB \cdot \tan(76^\circ)
     \]
   - From point \( B \) (angle \( 74^\circ \)), we have:
     \[
     h = BC = AB \cdot \tan(74^\circ)
     \]

3. **Calculate Depth of the Canyon:**
   - The depth \( D \) of the canyon can be found by the height from both angles:
     \[
     D = AB \cdot (\tan(76^\circ) - \tan(74^\circ))
     \]
   - Substituting \( AB = 5035 \):
     \[
     D = 5035 \cdot (\tan(76^\circ) - \tan(74^\circ))
     \]

4. **Calculate Values:**
   - Using a calculator:
     - \( \tan(76^\circ) \approx 4.010 \)
     - \( \tan(74^\circ) \approx 3.487 \)
   - Therefore:
     \[
     D \approx 5035 \cdot (4.010 - 3.487) \approx 5035 \cdot 0.523 \approx 2634.88 \text{ ft}
     \]

### Conclusion:
The depth of the canyon is approximately **2635 ft**.

Would you like further details on this solution or have any questions? Here are some related questions you might consider:

1. How does the angle of elevation affect the calculation of height?
2. What would happen to the depth if the length of the bridge were shorter?
3. How do you determine the angles of elevation in practical scenarios?
4. Can you explain how to use different trigonometric functions to find distances?
5. What are the real-life applications of this type of calculation?

**Tip:** Always ensure your calculator is set to the correct mode (degrees/radians) when working with angles.

Discover how to calculate the depth of a canyon based on a bridge length of 5035 ft and angles of elevation of 76 and 74 degrees. This detailed solution employs trigonometric relationships and provides step-by-step guidance suitable for high school students.

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