Explain how to solve the following question.

Surveying: From a point A hat is 10 meters above level ground, the angle of elevation of the top of a building is 42 degrees and the angle of depression of the base of the building is 8 degrees. Approximate the height of the building.

To solve this question, we can use trigonometric concepts and equations. Here's how:

Step 1: Draw a diagram
Draw a diagram representing the situation described in the question. Label the point A as the observer's position and mark a perpendicular line from point A to the ground to represent the height of 10 meters.

Step 2: Identify the angle of elevation and angle of depression
As per the question, the angle of elevation of the top of the building is 42 degrees, and the angle of depression of the base of the building is 8 degrees. Mark these angles on the diagram.

Step 3: Identify relevant trigonometric ratios
Using trigonometry, we can relate the angles and sides of a right triangle. In this case, we need to consider the tangent function (tan) because we have the opposite and adjacent sides.

- For the angle of elevation (42 degrees):
The tangent of the angle of elevation is equal to the height of the building divided by the distance from the observer to the building.

- For the angle of depression (8 degrees):
The tangent of the angle of depression is equal to the height of the building divided by the distance from the observer to the building.

Step 4: Set up the trigonometric equations
Let's denote the height of the building as 'h' and the distance from the observer to the building as 'x'. Using the information from steps 3 and the given 10 meters height, we can set up the following equations:

1) tan(42 degrees) = h / x
2) tan(8 degrees) = h / (x + 10)

Step 5: Solve the equations for the height of the building
Now, we have two equations with two unknowns (h and x). To solve for the height, we can use substitution or elimination methods.

Using the first equation, isolate 'h' by multiplying 'x' to both sides:
h = x * tan(42 degrees)

Substitute this value of 'h' into the second equation:
tan(8 degrees) = (x * tan(42 degrees)) / (x + 10)

Rearrange the equation to solve for 'x':
(x + 10) * tan(8 degrees) = x * tan(42 degrees)

Expand and rearrange:
x * tan(8 degrees) + 10 * tan(8 degrees) = x * tan(42 degrees)

Now, isolate 'x' by moving all the 'x' terms to one side:
x * (tan(8 degrees) - tan(42 degrees)) = 10 * tan(8 degrees)

Finally, solve for 'x':
x = (10 * tan(8 degrees)) / (tan(8 degrees) - tan(42 degrees))

After calculating the value of 'x', substitute it back into any of the original equations to find the height 'h' of the building:
h = x * tan(42 degrees)

Approximate the height of the building with appropriate units based on the calculations.

Remember, in any step, you can use a scientific calculator to find trigonometric functions like tangent (tan) and substitute the given angle values to make the calculations easier.