From her eye, which stands 1.68 meters above the ground, Hannah measures the angle of elevation to the top of a prominent skyscraper to be 31degrees 
∘
 . If she is standing at a horizontal distance of 194 meters from the base of the skyscraper, what is the height of the skyscraper? Round your answer to the nearest hundredth of a meter if necessary.

To solve this problem, we can use trigonometry. Specifically, the **tangent** function relates the angle of elevation, the opposite side (the height of the skyscraper above Hannah's eye level), and the adjacent side (the horizontal distance to the base of the skyscraper).

### Step-by-Step Solution:

1. **Define the variables**:
   - \( h \): the height of the skyscraper above Hannah's eye level.
   - The total height of the skyscraper will be \( h + 1.68 \), since Hannah's eye is 1.68 meters above the ground.
   - \( d = 194 \) meters is the horizontal distance from Hannah to the base of the skyscraper.
   - The angle of elevation \( \theta = 31^\circ \).

2. **Tangent function**:
   \[
   \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{d}
   \]

   Plug in the values:
   \[
   \tan(31^\circ) = \frac{h}{194}
   \]

3. **Solve for \( h \)**:
   \[
   h = 194 \times \tan(31^\circ)
   \]

4. **Calculate the value of \( h \)**:
   Now, we'll calculate this value using a calculator.

\[
h = 194 \times \tan(31^\circ) \approx 194 \times 0.6009 \approx 116.58 \, \text{meters}
\]

5. **Total height of the skyscraper**:
   The total height of the skyscraper is:
   \[
   \text{Total height} = h + 1.68 = 116.58 + 1.68 = 118.26 \, \text{meters}
   \]

### Final Answer:
The height of the skyscraper is approximately **118.26 meters**.

Would you like further details or have any questions?

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Here are 5 related questions to expand on this concept:

1. What trigonometric functions are used for angles of depression?
2. How would the answer change if Hannah was further from the skyscraper?
3. Can the tangent function be used for angles greater than \(90^\circ\)?
4. What would happen if Hannah measured the angle of elevation from a higher position?
5. How can we calculate the distance from the base of the building if the height and angle are known?

**Tip**: When using a scientific calculator, ensure it's set to degrees (°) when working with angles like 31°.

From her eye, which stands 1.68 meters above the ground, Hannah measures the angle of elevation to the top of a prominent skyscraper to be 31 degrees. If she is standing at a horizontal distance of 194 meters from the base of the skyscraper, what is the height of the skyscraper? Round your answer to the nearest hundredth of a meter if necessary.

Learn how to calculate the height of a skyscraper using the angle of elevation and horizontal distance with the tangent function. This step-by-step solution involves key trigonometric concepts and is suitable for students in Grades 9-12.

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