Answer :
To solve this problem, we will use basic trigonometric concepts. Specifically, we will work with the angle of depression and the tangent function.
Step-by-step Solution:
Understand the Scenario:
- Ted's eye level is 16 feet above the ground.
- The angle of depression to the dog is 40 degrees.
- We need to find the horizontal distance that the dog must walk to reach the spot directly below Ted. This horizontal distance will be the base of a right triangle, where the vertical side (opposite to the angle of depression) is 16 feet.
Relate the Angle of Depression to the Scenario:
- The angle of depression from a point above an object to the object can be visualized as the angle between the horizontal line at the observer's level and the line of sight to the object.
- By the properties of parallel lines and transversal, the angle of depression of 40 degrees is equal to the angle of elevation from the dog to Ted's eyes.
Use Trigonometry to Find the Distance:
- In the right triangle formed, the side opposite to the 40 degree angle is 16 feet (height of Ted's eyes from the ground).
- We want to find the side adjacent to the 40 degree angle (the distance the dog must walk).
- We'll use the tangent function, which relates the opposite and adjacent sides of a right triangle:
[tex]\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}[/tex]
In this case:
[tex]\tan(40^\circ) = \frac{16}{d}[/tex]Solve for [tex]d[/tex]:
- Rearrange the formula to solve for [tex]d[/tex]:
[tex]d = \frac{16}{\tan(40^\circ)}[/tex]
Calculate the Exact Value:
- Using a calculator:
[tex]d \approx \frac{16}{0.8391} \approx 19.1\text{ feet}[/tex]
- Using a calculator:
Therefore, the dog must walk approximately 19.1 feet to reach the spot directly below Ted's position in the barn.
This problem demonstrates the practical application of trigonometric functions to solve real-world problems. Understanding angles of depression and elevation can be very useful in many fields including engineering, architecture, and even aviation.
