https://file.aiquickdraw.com/mathbot/file/172546021875192ahtvuk.jpg

The problem in the image involves a right triangle with a ladder leaning against a wall, forming the hypotenuse of the triangle. The task is to find the angle of elevation (\(\theta\)) from the ground to the top of the ladder.

### Problem Summary:
- **Ladder length** (hypotenuse): \(15\) feet.
- **Height** reached by the ladder on the wall (opposite side): \(9\) feet.

We are required to find the angle \(\theta\), which is the angle of elevation.

### Solution:
To find the angle \(\theta\), we use the sine function, which is defined as:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
Substitute the given values:
\[
\sin(\theta) = \frac{9}{15} = 0.6
\]

Now, to find the angle \(\theta\), take the inverse sine (\(\sin^{-1}\)) of 0.6:
\[
\theta = \sin^{-1}(0.6)
\]

Using a calculator:
\[
\theta \approx 36.87^\circ
\]

### Final Answer:
The angle of elevation \(\theta\) is approximately \(36.87^\circ\).

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Do you want details or have any questions?

Here are some related questions you might find interesting:
1. How would the angle change if the ladder were longer but the height remained the same?
2. What would be the angle of elevation if the ladder reached a height of 12 feet instead of 9 feet?
3. How would you calculate the length of the ladder if the angle and height are known?
4. What is the significance of the angle of elevation in real-life applications like this scenario?
5. Can you determine the base length of the triangle if the ladder's angle of elevation is known?

**Tip:** Remember, the trigonometric functions can help solve many problems involving right triangles.

Let θ be the angle of elevation from a point where you stand to the top of a cliff. If the height of the cliff is 9 feet and the hypotenuse of the triangle is 15 feet, find θ.

Learn how to calculate the angle of elevation when a ladder reaches 9 feet high on a wall with a length of 15 feet. This problem uses trigonometric concepts and is suitable for students in Grades 8-10.

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