Suppose that you are headed toward a plateau 30.5
 
meters high.  If the angle of elevation to the top of the plateau is 18.5
degrees​, 
how far are you from the base of the​ plateau?

To solve this problem, we can use trigonometry, specifically the tangent function, which relates the angle of elevation to the opposite side and adjacent side of a right triangle. 

The formula is:

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]

Where:
- \(\theta = 18.5^\circ\) is the angle of elevation,
- the opposite side is the height of the plateau, \(30.5\) meters,
- the adjacent side is the distance from the base of the plateau, which we need to find.

We can rearrange the formula to solve for the adjacent side:

\[
\text{adjacent} = \frac{\text{opposite}}{\tan(\theta)}
\]

Substitute the given values:

\[
\text{adjacent} = \frac{30.5}{\tan(18.5^\circ)}
\]

Now, let's calculate the distance.

\[
\text{adjacent} = \frac{30.5}{\tan(18.5^\circ)} \approx \frac{30.5}{0.3336} \approx 91.4 \text{ meters}
\]

So, the distance from the base of the plateau is approximately **91.4 meters**.

Would you like further details or explanations on any step?

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Here are 5 related questions to explore:
1. How does the angle of elevation affect the distance calculation in this problem?
2. What other trigonometric functions could be used to solve this problem?
3. How would the result change if the height of the plateau were different?
4. How can we verify the correctness of this calculation using different trigonometric approaches?
5. What would happen if the angle of elevation were larger, say 30 degrees?

**Tip:** Remember that angles of elevation are measured from the horizontal, so they are always between 0° and 90° for practical applications like this!

Suppose that you are headed toward a plateau 30.5 meters high. If the angle of elevation to the top of the plateau is 18.5 degrees, how far are you from the base of the plateau?

Solve this trigonometry problem involving the angle of elevation and the height of a plateau. Using the tangent function, calculate the distance from the base of a plateau 30.5 meters high with an angle of elevation of 18.5°. Suitable for students in Grades 8-10.

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