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The problem presented in the image involves determining the height of a building using two angles of elevation, measured from two distinct points 100 meters apart. The angles given are 33° and 46°.

This is a classic trigonometry problem that can be solved using the tangent function. Let's break it down:

### Known Values:
- Distance between two points: 100 meters.
- First angle of elevation (from farther point): 33°.
- Second angle of elevation (from closer point): 46°.
- Height of the building: \( h \) (to be determined).

### Steps to Solve:

1. Let’s define two right triangles formed by the two measurement points and the height of the building. The point closest to the building will give the tangent equation for the 46° angle, and the point farther away will give the equation for the 33° angle.

2. Let \( d \) be the distance from the point closer to the building. Then, the distance from the farther point will be \( d + 100 \) meters.

#### Equation for the closer point:
Using the tangent of the 46° angle:
\[
\tan(46^\circ) = \frac{h}{d}
\]
\[
h = d \cdot \tan(46^\circ)
\]

#### Equation for the farther point:
Using the tangent of the 33° angle:
\[
\tan(33^\circ) = \frac{h}{d + 100}
\]
\[
h = (d + 100) \cdot \tan(33^\circ)
\]

3. Set these two expressions for \( h \) equal to each other:
\[
d \cdot \tan(46^\circ) = (d + 100) \cdot \tan(33^\circ)
\]

4. Now, solve for \( d \):
\[
d \cdot \tan(46^\circ) = d \cdot \tan(33^\circ) + 100 \cdot \tan(33^\circ)
\]
\[
d \cdot (\tan(46^\circ) - \tan(33^\circ)) = 100 \cdot \tan(33^\circ)
\]
\[
d = \frac{100 \cdot \tan(33^\circ)}{\tan(46^\circ) - \tan(33^\circ)}
\]

5. After calculating \( d \), substitute it into the equation for \( h \) to find the height of the building.

Let me calculate this for you.The height of the building, based on the given angles of elevation (33° and 46°) and the 100-meter distance between the two points, is approximately **174 meters**.

Thus, the correct answer is **174m**.

Would you like further details on any step, or do you have any additional questions?

Here are 5 related questions:
1. How can trigonometric functions be applied to other real-life problems?
2. How would the solution change if the distance between the two points was different?
3. What are the key differences between solving with sine, cosine, and tangent in such problems?
4. How can we verify the accuracy of the solution using a different method?
5. What would the angles need to be if the height of the building were doubled?

**Tip:** Always double-check the angle mode on your calculator (degrees vs. radians) when performing trigonometric calculations.

Para medir la altura de un edificio se miden los ángulos de elevación desde dos puntos distantes 100m. ¿Cuál es la altura si los ángulos son de 33° y 46°?

Learn how to calculate the height of a building using the angles of elevation (33° and 46°) measured from two points 100 meters apart. This problem involves applying trigonometric functions and solving for the unknown height using tangent formulas. Suitable for high school students.

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