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

To solve this problem, we need to use trigonometry. Let's break down the problem step-by-step:

1. **Understanding the Problem:**
   - We have a pole \(PQ\) and a building \(BC\).
   - The building \(BC\) is 30 feet tall.
   - From points \(P\) and \(Q\), the angles of elevation to the top of the building (point \(B\)) are \(30^\circ\) and \(40^\circ\), respectively.
   - We want to find the height of the pole \(PQ\).

2. **Define the Positions and Variables:**
   - Let the distance from point \(P\) to the base of the building \(C\) be \(x\) feet.
   - Let the distance from point \(Q\) to the base of the building \(C\) be \(y\) feet.
   - Since \(Q\) is closer to \(C\) than \(P\), we have \(x > y\).
   - Let the height of the pole \(PQ\) be \(h\) feet.

3. **Apply the Tangent Trigonometric Formula:**
   - For point \(P\):
     \[
     \tan(30^\circ) = \frac{\text{height of building}}{\text{distance from } P \text{ to } C} = \frac{30}{x}
     \]
     Solving for \(x\):
     \[
     x = \frac{30}{\tan(30^\circ)}
     \]

   - For point \(Q\):
     \[
     \tan(40^\circ) = \frac{\text{height of building}}{\text{distance from } Q \text{ to } C} = \frac{30}{y}
     \]
     Solving for \(y\):
     \[
     y = \frac{30}{\tan(40^\circ)}
     \]

4. **Calculate the Height of the Pole \(PQ\):**
   The height of the pole can be found by considering the right triangle formed by the pole, its shadow, and the line from the base of the pole to the top of the pole.

Now let's calculate these values.The calculated distances from points \(P\) and \(Q\) to the base of the building \(C\) are:

- Distance \(x\) from \(P\) to \(C\): approximately \(51.96\) feet.
- Distance \(y\) from \(Q\) to \(C\): approximately \(35.75\) feet.

### 5. Finding the Height of the Pole \(PQ\):

Since the pole's height creates a triangle with the base, we can use a similar method to find the height of the pole. The angles and distances will relate in a similar trigonometric way.

The difference between the distances from \(P\) and \(Q\) gives us the difference in the height observed from these two points. Using the tangent relation again:

\[
\tan(30^\circ) = \frac{h}{x - y}
\]

Let's solve for \(h\).The height of the pole \(PQ\) is approximately **9.36 feet**.

### Summary:
- The building is 30 feet tall.
- The distances from points \(P\) and \(Q\) to the base of the building \(C\) are approximately 51.96 feet and 35.75 feet, respectively.
- The height of the pole \(PQ\) is approximately 9.36 feet.

Would you like more details or have any questions? Here are some related questions that might interest you:

1. How would the problem change if the angles of elevation were different?
2. What is the formula for the angle of depression in trigonometry?
3. How can we calculate the distance between two points using trigonometry?
4. How do we solve problems involving the shadow of a pole and its height?
5. What is the relationship between trigonometric functions and right triangles?

**Tip:** Understanding how to use trigonometric functions like tangent, sine, and cosine in different contexts is key to solving various real-world problems!

Learn how to calculate the height of a pole using trigonometry and angles of elevation. Given a 30 feet tall building and angles of elevation from points P and Q to the top of the building, find the height of the pole PQ. Includes step-by-step solution using trigonometric functions.

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