From a horizontal distance of 8.5 km, a pilot observes that the angles of depression of the top and the base of a control tower are 30° and 40° respectively. Calculate, correct to three significant figures :
(a) the shortest distance between the pilot and the base of the control tower;
(b) the height of the control tower.

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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From a horizontal distance of 8.5 km, a pilot observes that the angles of depression of the top and the base of a control tower are 30° and 40° respectively. Calculate, correct to three significant figures: (a) the shortest distance between the pilot and the base of the control tower; (b) the height of the control tower.

Solve a trigonometry problem involving angles of depression observed by a pilot from a horizontal distance of 8.5 km. Learn how to calculate the shortest distance to the base of a control tower and its height using trigonometric formulas. Suitable for Grades 10-12.

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