To determine the distance of the boat from the shoreline, we can follow these steps:
1. Understand the Problem:
- The height of the observation tower is 60 feet.
- The angle of depression from the tower to the boat is 4.5 degrees.
2. Visualize the Scenario:
- Imagine a right triangle where the height of the tower forms the vertical leg, the distance from the base of the tower to the point where the line of sight meets the water forms the horizontal leg, and the line of sight from the top of the tower to the boat forms the hypotenuse.
- The angle of depression is the angle between the horizontal line at the level of the observer's eyes and the line of sight to the boat.
3. Establish the Trigonometric Relationship:
- The angle of depression and the corresponding angle within the right triangle are equal because they are alternate interior angles.
- The vertical leg (opposite side) of the triangle is the height of the tower, which is 60 feet.
- The horizontal leg (adjacent side) is the distance we need to find.
4. Apply the Tangent Function:
- The tangent of an angle in a right triangle relates the opposite side to the adjacent side:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
- In this case:
[tex]\[
\tan(4.5^\circ) = \frac{\text{height of the tower}}{\text{distance from the shoreline}}
\][/tex]
5. Setup the Equation:
- Let [tex]\(d\)[/tex] represent the distance from the shoreline to the boat:
[tex]\[
\tan(4.5^\circ) = \frac{60}{d}
\][/tex]
6. Solve for d:
- Rearrange the equation to solve for [tex]\(d\)[/tex]:
[tex]\[
d = \frac{60}{\tan(4.5^\circ)}
\][/tex]
7. Result:
- By substituting the values into the equation and calculating, the distance [tex]\(d\)[/tex] is approximately:
[tex]\[
d \approx 762.3722841704823
\][/tex]
Therefore, the boat is approximately 762.37 feet from the shoreline.
