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The problem is about finding the height of an antenna located on top of a building, given two angles of elevation and the distance from the base. Let's outline and solve it step by step:

### Problem Breakdown:
1. **Distance from the base**: \( 300 \, \text{feet} \)
2. **Angle of elevation to the top of the building**: \( 40^\circ \)
3. **Angle of elevation to the top of the antenna**: \( 43^\circ \)

### Variables:
- \( h_b \): height of the building
- \( h_a \): height of the antenna
- Total height: \( h_b + h_a \)

### Step 1: Use the angle of elevation to the top of the building
We can use the tangent function to find \( h_b \), the height of the building:
\[
\tan(40^\circ) = \frac{h_b}{300}
\]
\[
h_b = 300 \cdot \tan(40^\circ)
\]

### Step 2: Use the angle of elevation to the top of the antenna
Similarly, use the tangent function to find the total height (building + antenna):
\[
\tan(43^\circ) = \frac{h_b + h_a}{300}
\]
\[
h_b + h_a = 300 \cdot \tan(43^\circ)
\]

### Step 3: Solve for \( h_a \)
To find the height of the antenna, subtract \( h_b \) from the total height:
\[
h_a = 300 \cdot \tan(43^\circ) - 300 \cdot \tan(40^\circ)
\]

Let me calculate that for you.The height of the building is approximately \( 251.73 \, \text{feet} \), and the height of the antenna on top of the building is approximately \( 28.02 \, \text{feet} \).

Would you like more details on the process or have any further questions?

Here are 5 related questions:
1. How does the tangent function relate angles and distances in right triangles?
2. What other trigonometric functions can be used to solve height problems?
3. How would the problem change if the observation point was closer to the building?
4. Can you solve this problem using sine or cosine instead of tangent?
5. What happens to the height calculations if the angles of elevation are measured incorrectly?

**Tip**: Always check your angle mode (degrees vs. radians) when using trigonometric functions in calculators!

There is an antenna on the top of a building. From a location 300 feet from the base of the building, the angle of elevation to the top of the building is measured to be 40°. From the same location, the angle of elevation to the top of the antenna is measured to be 43°. Find the height of the antenna.

This problem demonstrates how to calculate the height of an antenna on top of a building using angles of elevation and trigonometric functions. Learn how to solve it step-by-step, including key formulas like tan(angle) = opposite / adjacent. Suitable for Grades 10-12.

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