PQ is a pole and BC is a 30 feet tall building .from Pand Q the angkes if elevation of B are 30° and 45° respectively what is the height of the pole in feet take √3=1.732

To solve this problem, we'll use trigonometry to find the height of the pole (PQ). The scenario involves two angles of elevation, one from point P and one from point Q, to the top of the building BC, which is 30 feet tall.

Let's define the variables:
- Let the height of the pole \( PQ = h \).
- The distance between the pole and the building is divided into two segments: the distance from P to B (denoted as \( d_1 \)) and the distance from Q to B (denoted as \( d_2 \)).

From the problem, we know:
1. From point P, the angle of elevation to point B (the top of the building) is \( 30^\circ \).
2. From point Q, the angle of elevation to point B is \( 45^\circ \).

### Step 1: Express the height of the building using trigonometry

From the first angle of elevation at point P:
\[
\tan(30^\circ) = \frac{\text{Height of the building}}{d_1} = \frac{30}{d_1}
\]
Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \approx 0.577 \), we get:
\[
0.577 = \frac{30}{d_1}
\]
Solving for \( d_1 \):
\[
d_1 = \frac{30}{0.577} \approx 51.97 \text{ feet}
\]

### Step 2: Express the height of the building using the second angle

From the second angle of elevation at point Q:
\[
\tan(45^\circ) = \frac{\text{Height of the building}}{d_2} = \frac{30}{d_2}
\]
Since \( \tan(45^\circ) = 1 \), we get:
\[
d_2 = 30 \text{ feet}
\]

### Step 3: Determine the height of the pole

The total distance between P and Q is:
\[
\text{Distance between P and Q} = d_1 - d_2 = 51.97 - 30 = 21.97 \text{ feet}
\]

Therefore, the height of the pole is approximately 21.97 feet.

Would you like further details on any part of this solution?

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Here are five related questions:
1. How does the angle of elevation affect the height calculation?
2. What happens to the height calculation if the angle of elevation increases?
3. How would you find the height of the building if only one angle of elevation was provided?
4. What is the relationship between the distance from the object and the angle of elevation?
5. How would the problem change if the height of the building was unknown?

**Tip:** Always double-check your trigonometric ratios and ensure your angles are correctly calculated for accurate results!

Learn how to calculate the height of a pole using angles of elevation in trigonometry. Solve a problem where angles of 30° and 45° from points P and Q to a 30 feet tall building determine the pole's height. Detailed step-by-step solution provided.

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