https://file.aiquickdraw.com/mathbot/file/1725963325761r0cc82zb.jpg

The image shows a right triangle formed by a building, its shadow, and the sunlight, where:

- The shadow length (adjacent side, \( h_a \)) is 67.2 meters.
- The angle \( \theta \) between the sun's rays and the ground is \( 50^\circ \).
- We are asked to determine the height of the building (opposite side, \( h_o \)) and the length of the hypotenuse.

### 1. Determine the height of the building:
We can use trigonometric functions. Specifically, the tangent of the angle \( \theta \) relates the opposite side (height of the building, \( h_o \)) to the adjacent side (shadow length, \( h_a \)):

\[
\tan(\theta) = \frac{h_o}{h_a}
\]

Solving for \( h_o \):

\[
h_o = h_a \cdot \tan(\theta)
\]

Substituting the given values:

\[
h_o = 67.2 \cdot \tan(50^\circ)
\]

Now, let's calculate this.The height of the building is approximately \( 80.09 \, \text{m} \).

### 2. Determine the hypotenuse of the triangle:
We can use the Pythagorean Theorem to find the hypotenuse \( h \), which states:

\[
h^2 = h_o^2 + h_a^2
\]

Substituting the known values:

\[
h = \sqrt{h_o^2 + h_a^2}
\]

Let's calculate the hypotenuse next.It seems like I can’t do more advanced data analysis right now. Please try again later.

Let me guide you on how to manually calculate the hypotenuse:

You can use the Pythagorean theorem formula:

\[
h = \sqrt{h_o^2 + h_a^2}
\]

Where:
- \( h_o = 80.09 \, \text{m} \) (height of the building)
- \( h_a = 67.2 \, \text{m} \) (length of the shadow)

Substitute the values and solve:

\[
h = \sqrt{(80.09)^2 + (67.2)^2}
\]

This will give you the hypotenuse value.

Would you like to clarify anything, or need further details?

Here are some follow-up questions:
1. How is the tangent function used in solving right triangles?
2. What are the other trigonometric ratios for right triangles?
3. Why does the Pythagorean theorem apply here?
4. Can you derive the shadow length given the building height?
5. How would the answer change if the angle was steeper (closer to 90°)?

**Tip**: The tangent ratio is particularly useful when you know the adjacent side and angle but need the opposite side.

On a sunny day, a tall building casts a shadow that is 67.2 m long. The angle between the sun's rays and the ground is θ = 50.0°, as the image shows. Determine the height of the building.

Learn how to calculate the height of a building and the hypotenuse of a right triangle formed by its shadow using trigonometry. Given a shadow length of 67.2 meters and an angle of 50°, apply the tangent function and Pythagorean theorem to solve this problem.

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