a bridge is being built across a canyon. the length of the bridge is 5065 ft. from the deepest point in the canyon, the angles of elevation of the ends of the bridge are 77 degrees and 72 degrees. how deep is the canyon?

To find the depth of the canyon, we can use trigonometric relationships involving the angles of elevation and the lengths involved. Let's denote:

- \( A \) as the point directly below the observer at the deepest point in the canyon.
- \( B \) and \( C \) as the ends of the bridge, with \( AB = AC = x \) (the horizontal distances from \( A \) to \( B \) and \( C \)).
- The angle of elevation to \( B \) is \( 77^\circ \) and to \( C \) is \( 72^\circ \).

Using the tangent function, we have:

\[
h = x \tan(77^\circ)
\]
\[
h = (5065 - x) \tan(72^\circ)
\]

Setting the two expressions for \( h \) equal to each other gives us:

\[
x \tan(77^\circ) = (5065 - x) \tan(72^\circ)
\]

Now we can solve for \( x \):

\[
x \tan(77^\circ) + x \tan(72^\circ) = 5065 \tan(72^\circ
\]
\[
x(\tan(77^\circ) + \tan(72^\circ)) = 5065 \tan(72^\circ)
\]
\[
x = \frac{5065 \tan(72^\circ)}{\tan(77^\circ) + \tan(72^\circ)}
\]

Calculating the values for \( x \) and then substituting back to find \( h \):

1. Calculate \( \tan(77^\circ) \) and \( \tan(72^\circ) \).
2. Solve for \( x \).
3. Substitute \( x \) into either equation to find \( h \).

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A bridge is being built across a canyon. The length of the bridge is 5065 ft. From the deepest point in the canyon, the angles of elevation of the ends of the bridge are 77 degrees and 72 degrees. How deep is the canyon?

Explore how to calculate the depth of a canyon using trigonometric relationships with a bridge length of 5065 ft and angles of elevation of 77° and 72°. This problem is suitable for high school students learning about trigonometry.

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