To solve this problem, we can use trigonometric relationships, specifically the tangent function, which relates the angle of depression to the height and distance from the base of the tower.
Given:
- Height of the tower (h) = 20 meters
- Angle of depression (θ) = 30°
From the top of the tower, the angle of depression to the boat means that if we draw a horizontal line from the top of the tower, the angle between this line and the line of sight to the boat is 30°.
Let \( d \) be the horizontal distance from the foot of the tower to the boat. We can use the tangent of the angle of depression, which is also the angle from the boat's perspective looking up toward the tower:
\[ \tan(θ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{d} \]
In this case, h is the height of the tower and \( d \) is the distance from the boat to the foot of the tower.
Plugging in the values we have:
\[ \tan(30°) = \frac{20}{d} \]
We know that \( \tan(30°) = \frac{1}{\sqrt{3}} \) or approximately 0.577.
So we can write:
\[ \frac{1}{\sqrt{3}} = \frac{20}{d} \]
To find \( d \), we can cross-multiply:
\[ d = 20 \sqrt{3} \]
Now, we can find the numeric value of \( d \):
\[ d \approx 20 \times 1.732 \approx 34.64 \text{ meters} \]
Thus, the distance from the boat to the foot of the tower is approximately 34.64 meters.