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Let's break down the problem to provide a detailed solution. I'll solve each part:

### Part a: Draw a diagram
- Imagine a person standing on a building that is 33 meters high.
- The person looks across the road to a church.
- The angle of depression to the bottom of the church is \( 25^\circ \).
- The angle of elevation to the top of the church is \( 18^\circ \).

In the diagram:
- Label the height of the building as 33 m.
- Draw a horizontal line representing the person's line of sight (level with the top of the building).
- Draw two lines from the top of the building:
  - One at a \( 25^\circ \) angle downwards to the base of the church.
  - One at an \( 18^\circ \) angle upwards to the top of the church.

Next, let's move on to the calculation.

### Part b: Calculate the height of the church
We need to find the height of the church, so we will break it down into two components: 
1. The horizontal distance from the building to the church.
2. The height of the church itself.

#### Step 1: Use trigonometry to find the horizontal distance.

Let:
- \( h_b = 33 \, \text{m} \) be the height of the building.
- \( \theta_d = 25^\circ \) is the angle of depression to the base of the church.

From the angle of depression:
\[
\tan(25^\circ) = \frac{33}{d}
\]
Solving for \( d \) (the horizontal distance):
\[
d = \frac{33}{\tan(25^\circ)}
\]

#### Step 2: Use the angle of elevation to find the additional height of the church.

Let:
- \( \theta_e = 18^\circ \) be the angle of elevation to the top of the church.
- \( h_c \) be the height of the church.

Using trigonometry again:
\[
\tan(18^\circ) = \frac{h_c - 33}{d}
\]
We already know \( d \), so now we solve for \( h_c \):
\[
h_c = 33 + d \cdot \tan(18^\circ)
\]

Let me compute the values now.1. The horizontal distance between the building and the church is approximately \( 70.77 \, \text{m} \).
2. The total height of the church is approximately \( 55.99 \, \text{m} \).

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

Here are 5 related questions to expand on this topic:
1. How do we apply trigonometry in real-world problems involving angles of depression and elevation?
2. What is the relationship between angles and the horizontal distance in trigonometric problems?
3. How does the angle of depression differ from the angle of elevation?
4. Can you solve a similar problem but with different angles or heights?
5. How would the result change if the building height or the angles were altered?

**Tip:** In trigonometry, always ensure that your calculator is set to the correct mode (degrees or radians) based on the problem.

A person on top of a 33 m high building looks across the road to a church on the other side. The angle of depression of the bottom of the church is 25°, and the angle of elevation of the top of the church is 18°. Draw a diagram to display this information and calculate the height of the church.

Learn to apply trigonometry to calculate the height of a structure from a given distance and angles of elevation and depression, using a 33 m high building and a church as examples. This problem teaches the use of tangent ratios and the importance of angle measurements.

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