A survey team is trying to estimate the height of a mountain above a level plain. From one point on the plain, they observe that the angle of elevation to the top of the mountain is 24 degrees. From a point 3000 feet closer to the mountain along the plain, they find that the angle of elevation is 29 degrees. How high (in feet) is the mountain?

This problem can be solved using trigonometry, specifically the concept of right triangles and the tangent function. Let's break down the problem step by step:

### Step 1: Understanding the Setup

- Let \( h \) be the height of the mountain.
- Let \( x \) be the distance from the first point of observation to the base of the mountain.
- At the first point, the angle of elevation is 24 degrees.
- At the second point, 3000 feet closer to the mountain, the angle of elevation is 29 degrees.

### Step 2: Set up the Trigonometric Equations

From the tangent function, we know that:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
where:
- \( \theta \) is the angle of elevation,
- "opposite" is the height of the mountain \( h \),
- "adjacent" is the distance along the ground from the point of observation to the base of the mountain.

We can now set up two equations based on the tangent function for the two observation points.

#### For the first point:
\[
\tan(24^\circ) = \frac{h}{x}
\]
which gives:
\[
h = x \cdot \tan(24^\circ)
\]

#### For the second point (3000 feet closer):
The distance from this point to the base of the mountain is \( x - 3000 \), so:
\[
\tan(29^\circ) = \frac{h}{x - 3000}
\]
which gives:
\[
h = (x - 3000) \cdot \tan(29^\circ)
\]

### Step 3: Solve the System of Equations

Now, we have two expressions for \( h \):
1. \( h = x \cdot \tan(24^\circ) \)
2. \( h = (x - 3000) \cdot \tan(29^\circ) \)

Since both expressions represent the height of the mountain, we can set them equal to each other:
\[
x \cdot \tan(24^\circ) = (x - 3000) \cdot \tan(29^\circ)
\]

Now, we can solve for \( x \), the distance from the first point to the base of the mountain.

Let's calculate that next.It seems that the value of \( x \) is negative, which isn't physically meaningful in this context. This suggests that we made an error in interpreting the angles or the distances. Let me reassess the problem setup and recompute it.

We will approach the problem carefully again and compute the correct values.It appears that the negative result suggests we might need to recheck the sign of distances and angle setup or constraints within the problem. Let's rework and solve it carefully.

The setup with angles should correctly form the trigonometric relationship of the tangent of the two angles properly

Learn how to estimate the height of a mountain using trigonometry by solving for height with angles of elevation and tangent functions. This problem involves key concepts such as right triangles, tangent formulas, and solving systems of equations. Suitable for students in Grades 10-12.

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