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Let's break down the problem and solve it step by step.

We are given:

- Triangle \( ABC \) and triangle \( ADC \) are right-angled triangles.
- The angle of elevation from point \( B \) to the top of the pole \( A \) is \( 39^\circ \).
- The angle of elevation from point \( D \) to the top of the pole \( A \) is \( 42^\circ \).
- Points \( B \) and \( D \) are \( 20 \) m apart.
- The height of the pole is \( h \).

Let’s solve both parts of the question:

### Part (a): Expression for the length of \( AD \)

Using the tangent of the given angles in the right triangles \( ABC \) and \( ADC \), we can set up the following relations for \( h \) (the height of the pole):

1. In triangle \( ABC \):
   \[
   \tan(39^\circ) = \frac{h}{BC}
   \]
   So,
   \[
   BC = \frac{h}{\tan(39^\circ)}
   \]

2. In triangle \( ADC \):
   \[
   \tan(42^\circ) = \frac{h}{CD}
   \]
   So,
   \[
   CD = \frac{h}{\tan(42^\circ)}
   \]

Since \( B \) and \( D \) are \( 20 \) meters apart, we have the relation:
\[
BC + CD = 20
\]
Substituting the values of \( BC \) and \( CD \):
\[
\frac{h}{\tan(39^\circ)} + \frac{h}{\tan(42^\circ)} = 20
\]

This is the equation for the height \( h \) of the pole.

### Part (b): Find the height of the pole

We can now solve the equation for \( h \). Let’s compute the values of \( \tan(39^\circ) \) and \( \tan(42^\circ) \), and substitute them into the equation.

Let me calculate this for you.The height of the pole \( h \) is approximately \( 8.5 \) meters to one decimal place.

### Summary:
- The expression for the length of \( AD \) involves solving the equation:
  \[
  \frac{h}{\tan(39^\circ)} + \frac{h}{\tan(42^\circ)} = 20
  \]
- The height of the pole is approximately \( 8.5 \) meters.

Would you like further clarification or details on this solution?

Here are 5 related questions to expand your understanding:
1. How does the tangent function relate to angles in a right triangle?
2. How can we use the sine or cosine functions to solve similar problems?
3. What happens to the height \( h \) if the distance between points \( B \) and \( D \) changes?
4. How would the problem change if we were given angles of depression instead of elevation?
5. What is the general method for solving trigonometric problems involving multiple triangles?

**Tip:** When working with trigonometric functions, always ensure your calculator is set to the correct mode (degrees or radians) depending on the angles given.

The angle of elevation from point B to the top of a pole AC is 39° and the angle of elevation from D, on the other side of the pole, is 42°. B and D are 20 m apart. Find an expression for the length of AD and find the height of the pole.

Solve a trigonometry problem involving angles of elevation from points B and D to the top of a pole. Learn how to find an expression for the length of AD and the height of the pole using tangent formulas. Suitable for students in Grades 10-12.

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