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, which relates the angle of elevation to the height and horizontal distance.

Here’s the breakdown of the solution:

### Step 1: Understanding the triangle
- The angle of elevation is 31°, and Hannah is 194 meters away from the base of the skyscraper.
- We need to find the height of the skyscraper above her eye level. Let’s call this height \( h \).
- The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side:
  \[
  \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
  \]
  In this case:
  \[
  \tan(31^\circ) = \frac{h}{194}
  \]

### Step 2: Solve for \( h \)
\[
h = 194 \times \tan(31^\circ)
\]

Let’s calculate this value.

### Step 3: Total height of the skyscraper
The total height of the skyscraper is the sum of the height \( h \) and Hannah’s eye level, which is 1.68 meters.

\[
\text{Total height} = h + 1.68
\]

Now let's calculate this.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, if you want to calculate it manually, follow these steps:

1. Find \( h = 194 \times \tan(31^\circ) \) to get the height above Hannah's eye level.
2. Add 1.68 meters to account for her eye height to get the total height of the skyscraper.

Let me know if you have any questions, or if you'd like to work through these calculations together.

Here are some related questions:
1. How would the answer change if the angle of elevation was larger?
2. What trigonometric function would you use if you knew the total height and needed to find the distance?
3. How would the height change if Hannah stood closer or farther from the skyscraper?
4. What is the relationship between angle of elevation and horizontal distance for the same building?
5. How does the height change if the angle of elevation increases by 10 degrees?

**Tip**: Always double-check if the trigonometric functions you're using (e.g., sine, cosine, or tangent) match the sides of the triangle in your problem.

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°. 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 by applying trigonometry. With an angle of elevation of 31° and a horizontal distance of 194 meters, find the total height by using the tangent function and adding Hannah's eye level (1.68 meters). Suitable for students in Grades 9-11.

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