We can solve this problem using basic trigonometric relationships from right-angled triangles. Let's break down the problem:

Given Information:

• The horizontal distance from the pilot to the control tower is $8.5 \, \text{km}$.
• The angle of depression to the top of the control tower is $30^\circ$.
• The angle of depression to the base of the control tower is $40^\circ$.

Let:

• $D$ be the shortest distance between the pilot and the base of the control tower (hypotenuse of the larger triangle).
• $h$ be the height of the control tower.

We can now set up two right-angled triangles:

1. A triangle formed between the pilot, the top of the tower, and the point vertically below the pilot.
2. A triangle formed between the pilot, the base of the tower, and the same point.

Using the trigonometric function for tangent (as we have opposite and adjacent sides for both triangles):

Step 1: Calculate the height of the control tower

For the angle of depression of $30^\circ$, we have: $\tan 30^\circ = \frac{h}{8.5}$

Solving for $h$: $h = 8.5 \times \tan 30^\circ$

For the angle of depression of $40^\circ$, the total distance between the pilot and the base of the tower is: $\tan 40^\circ = \frac{H}{8.5}$

Where $H$ represents the entire height from the pilot to the base of the control tower. Hence, solving for $H$: $H = 8.5 \times \tan 40^\circ$

Now, the height of the control tower $h$ is the difference between $H$ and ( h .

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