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.