Suppose that you are headed toward a plateau 68 meters high.  If the angle of elevation to the top of the plateau is 20degrees​, how far are you from the base of the​ plateau?
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Part 1
What is your distance from the base of the​ plateau?
  
enter your response here meters  ​(Round to two decimal places as​ needed.)

To solve this, we can use trigonometry. Specifically, the tangent function, because we know the angle of elevation and the height of the plateau, and we are looking for the distance from the base (adjacent side).

The tangent of an angle is the ratio of the opposite side (height of the plateau) to the adjacent side (distance from the base).

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

Here:
- \(\theta = 20^\circ\)
- opposite = height of the plateau = 68 meters
- adjacent = distance from the base (what we are solving for)

We rearrange the formula to solve for the adjacent side:

\[
\text{adjacent} = \frac{\text{opposite}}{\tan(\theta)} = \frac{68}{\tan(20^\circ)}
\]

Now, let’s calculate the distance.The distance from the base of the plateau is approximately 186.83 meters.

Would you like further details or have any questions about the process? Here are 5 related questions to explore:

1. How would the distance change if the angle of elevation were 30 degrees instead of 20 degrees?
2. What is the formula for the sine function in a right triangle, and how would it apply here?
3. If you walked toward the plateau and halved your distance, what would the new angle of elevation be?
4. What if the height of the plateau were unknown? How would you determine it using trigonometry?
5. How do we apply the cosine function in real-world elevation problems?

**Tip:** Remember that trigonometric functions can help solve many real-world distance and height problems, especially when dealing with angles of elevation or depression.

Suppose that you are headed toward a plateau 68 meters high. If the angle of elevation to the top of the plateau is 20 degrees, how far are you from the base of the plateau? Round to two decimal places as needed.

Solve a real-world trigonometry problem where you calculate the distance to the base of a plateau 68 meters high, using an elevation angle of 20 degrees. Learn how to apply the tangent function and key trigonometric concepts. Suitable for Grades 9-12.

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