To solve this problem, we need to find the depth of the canyon, $h$, using trigonometry. Here’s the setup based on the information provided:

1. Given:

   • Length of the bridge: $5055$ ft.
   • Angles of elevation from the deepest point of the canyon to the ends of the bridge: $78^\circ$ and $71^\circ$.

2. Objective:

   • Find the depth $h$ of the canyon.

Solution Outline

The triangle in the diagram can be divided into two right triangles. Let’s designate:

• $A$ and $B$ as the endpoints of the bridge.
• $C$ as the point directly below the bridge at the deepest point of the canyon.

The depth $h$ is the vertical distance from point $C$ to the line segment $AB$ (the bridge).

Using Trigonometry

Since we have two right triangles, we can use the tangent function for each angle:

Step 1: Set up the equations for each side of the bridge

1. For the left side (angle $78^\circ$): $\tan(78^\circ) = \frac{h}{x}$ where $x$ is the horizontal distance from point $C$ to the left end of the bridge $A$.

   Rearranging, we get: $x = \frac{h}{\tan(78^\circ)}$

2. For the right side (angle $71^\circ$): $\tan(71^\circ) = \frac{h}{y}$ where $y$ is the horizontal distance from point $C$ to the right end of the bridge $B$.

   Rearranging, we get: $y = \frac{h}{\tan(71^\circ)}$

Step 2: Relate $x$ and $y$ to the total length of the bridge

Since $x + y = 5055$ ft, we substitute from the equations above: $\frac{h}{\tan(78^\circ)} + \frac{h}{\tan(71^\circ)} = 5055$

Step 3: Solve for $h$

1. Calculate $\tan(78^\circ)$ and $\tan(71^\circ)$:

   • $\tan(78^\circ) \approx 4.7046$
   • $\tan(71^\circ) \approx 2.9042$

2. Substitute these values into the equation: $\frac{h}{4.7046} + \frac{h}{2.9042} = 5055$

3. Combine terms: $h \left( \frac{1}{4.7046} + \frac{1}{2.9042} \right) = 5055$

4. Calculate the combined term: $h \times (0.2126 + 0.3444) = 5055$ $h \times 0.5570 = 5055$

5. Solve for $h$: $h = \frac{5055}{0.5570} \approx 9078 \text{ ft}$

Conclusion

The depth of the canyon is approximately 9078 feet.

Would you like further details, or do you have any questions?

Here are some related questions that could deepen your understanding:

1. How would the depth change if the length of the bridge was shorter?
2. What if the angles were both smaller, say $60^\circ$ and $55^\circ$?
3. How does the tangent function help determine heights in right triangles?
4. Could you solve this problem using the sine or cosine functions instead?
5. How would we set up the problem if the bridge was over a different shape, like an arc?

Tip: In trigonometry, the tangent of an angle is often used for finding heights when horizontal distances and angles are known, especially in right triangles.