To solve this problem, we can use trigonometric principles, specifically the tangent function, which relates the angle of a right triangle to the ratio of the side opposite the angle to the side adjacent to the angle.
Here are the step-by-step details:
1. Identify the Given Information:
- Height of the bird above sea level: [tex]\( h = 2 \)[/tex] meters
- Angle of depression to the fish: [tex]\( \theta = 15^\circ \)[/tex]
2. Understand the Relationship:
The angle of depression is the angle formed by the line of sight of the bird and the horizontal line when looking down at the fish. This forms a right triangle where:
- The height ([tex]\( h \)[/tex]) is the side opposite the angle [tex]\( \theta \)[/tex].
- The distance the bird must fly horizontally to be directly above the fish is the side adjacent to the angle [tex]\( \theta \)[/tex].
3. Use the Tangent Function:
The tangent function is defined as:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
In our case:
[tex]\[
\tan(15^\circ) = \frac{2 \text{ meters}}{\text{adjacent distance}}
\][/tex]
4. Solve for the Adjacent Distance:
Rearrange the tangent function to find the adjacent distance (which is the horizontal distance the bird must fly):
[tex]\[
\text{adjacent distance} = \frac{2 \text{ meters}}{\tan(15^\circ)}
\][/tex]
5. Calculate the Tangent of 15°:
Using a calculator, we find:
[tex]\[
\tan(15^\circ) \approx 0.2679
\][/tex]
6. Compute the Horizontal Distance:
[tex]\[
\text{adjacent distance} = \frac{2 \text{ meters}}{0.2679} \approx 7.4641 \text{ meters}
\][/tex]
7. Round to the Nearest Tenth:
The distance rounded to the nearest tenth is:
[tex]\[
7.5 \text{ meters}
\][/tex]
Therefore, the distance the bird must fly to be directly above the fish is approximately 7.5 meters.
