PQ is a pole and BC is a 30 feet tall building .from Pand Q the angkes if elevation of B are 30° and 40° respectively what is the height of the pole in feet

Let's solve the problem step by step by using the trigonometric concept of angles of elevation and right triangles.

### Problem Recap:
- **BC** is a 30-feet tall building.
- **PQ** is a pole with an unknown height, \(h\).
- From points **P** and **Q**, the angles of elevation to the top of the building (point **B**) are 30° and 40°, respectively.

### Solution:
1. **Setup and Diagram**: 
   - Assume point **P** is at a distance \(d\) from the base of the building **C**.
   - Point **Q** is at a distance \(d'\) from the base of the building **C**.
   - Both points **P** and **Q** lie on the ground.

2. **Using Trigonometry:**
   - For point **P**:
     \[
     \tan(30^\circ) = \frac{\text{height of building (BC)}}{\text{distance from P to building (PC)}}
     \]
     \[
     \tan(30^\circ) = \frac{30}{d}
     \]
     \[
     \frac{1}{\sqrt{3}} = \frac{30}{d} \quad (\text{since} \, \tan(30^\circ) = \frac{1}{\sqrt{3}})
     \]
     \[
     d = 30\sqrt{3}
     \]

   - For point **Q**:
     \[
     \tan(40^\circ) = \frac{30}{d'}
     \]
     \[
     d' = \frac{30}{\tan(40^\circ)}
     \]

3. **Height of the Pole (PQ):**
   - The height of the pole, \(h\), can be expressed as the difference between \(d\) and \(d'\) since the pole’s height difference from these points will be proportional to their distances.
   - Substituting the values, we get:
   
   \[
   h = d - d'
   \]

   Substituting \(d = 30\sqrt{3}\) and \(d' = \frac{30}{\tan(40^\circ)}\):

   \[
   h = 30\sqrt{3} - \frac{30}{\tan(40^\circ)}
   \]

   Calculate each term:

   \[
   \tan(40^\circ) \approx 0.8391
   \]
   
   So,

   \[
   d' \approx \frac{30}{0.8391} \approx 35.76
   \]

   And, \(d = 30\sqrt{3} \approx 51.96\)

   Finally,

   \[
   h \approx 51.96 - 35.76 \approx 16.20 \, \text{feet}
   \]

### Conclusion:

The height of the pole \(PQ\) is approximately **16.20 feet**.

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

Here are 5 related questions for further exploration:

1. How does changing the angle of elevation affect the calculated height?
2. What would be the height of the pole if both angles were the same?
3. How can this trigonometric approach be used to measure inaccessible heights?
4. What if the building height was unknown and needed to be calculated instead?
5. How does using different units (e.g., meters instead of feet) affect the trigonometric calculations?

**Tip:** Always draw a diagram when solving problems involving angles of elevation and height to visualize the trigonometric relationships more clearly.

Learn how to calculate the height of a pole using angles of elevation and trigonometric principles. Solve a problem where angles of elevation to a 30-feet tall building are given from two points, P and Q. Step-by-step solution provided.

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