PQ is a pole and BC is a 30 feet tall building .from Pand Q the angles 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, let's denote the following:

- \( h \) is the height of the pole \( PQ \) that we need to find.
- \( d_1 \) is the horizontal distance from point \( P \) to the base of the building \( BC \).
- \( d_2 \) is the horizontal distance from point \( Q \) to the base of the building \( BC \).

### Step 1: Analyze the information given
- \( BC = 30 \) feet (height of the building).
- Angle of elevation from \( P \) to \( B \) is \( 30^\circ \).
- Angle of elevation from \( Q \) to \( B \) is \( 45^\circ \).
- The distance between points \( P \) and \( Q \) is \( h \), the height of the pole.

### Step 2: Use trigonometry
From point \( P \), the tangent of the angle of elevation is given by:

\[
\tan 30^\circ = \frac{BC}{d_1} = \frac{30}{d_1}
\]

Since \( \tan 30^\circ = \frac{1}{\sqrt{3}} = \frac{1}{1.732} \), we have:

\[
\frac{1}{1.732} = \frac{30}{d_1}
\]

Solving for \( d_1 \):

\[
d_1 = 30 \times 1.732 = 51.96 \text{ feet}
\]

From point \( Q \), the tangent of the angle of elevation is given by:

\[
\tan 45^\circ = \frac{BC}{d_2} = \frac{30}{d_2}
\]

Since \( \tan 45^\circ = 1 \), we have:

\[
1 = \frac{30}{d_2}
\]

Solving for \( d_2 \):

\[
d_2 = 30 \text{ feet}
\]

### Step 3: Calculate the height of the pole \( PQ \)

The distance between points \( P \) and \( Q \) is equal to the height of the pole \( h \). This distance can also be expressed as the difference between \( d_1 \) and \( d_2 \):

\[
h = d_1 - d_2 = 51.96 - 30 = 21.96 \text{ feet}
\]

So, the height of the pole \( PQ \) is approximately \( \boxed{21.96} \) feet.

Would you like to see further details, or do you have any questions?

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**Relative Questions:**
1. What would be the height of the pole if the angles of elevation were 45° and 60° instead?
2. How would the problem change if the height of the building was doubled?
3. Can you derive a general formula for the height of the pole in terms of the given angles and building height?
4. What happens if the building's height and the angles are given in a different unit, such as meters and radians?
5. How would you calculate the height of the pole if the angles of elevation were reversed?

**Tip:** In trigonometry, angles of elevation and depression are often related to the tangent function, which can simplify calculations when heights and distances are involved.

Learn how to calculate the height of a pole using angles of elevation with a detailed step-by-step solution. This problem involves trigonometric concepts, including angles of elevation and the tangent function. Suitable for Grades 9-12.

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