Kara is zip-lining in the rainforest. She is standing at the top of Platform A ready to zip-line to Platform B. If the horizontal distance between the platforms is 500 feet and the length of the zip-line is 685 feet, find the angle of depression from Platform A to Platform B to the nearest tenth.
Kara is zip-lining in the rainforest. She is standing at the top of Platform A ready to zip-line to Platform B. If the horizontal distance between the platforms is 500 feet and the length of the zip-line is 685 feet, find the angle of depression from Platform A to Platform B to the nearest tenth.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Problem Statement
**Kara is zip-lining in the rainforest. She is standing at the top of Platform A ready to zip-line to Platform B. If the horizontal distance between the platforms is 500 feet and the length of the zip-line is 685 feet, find the angle of depression from Platform A to Platform B to the nearest tenth.**
### Solution
To find the angle of depression, we can use trigonometric relationships in the right triangle formed by the two platforms and the zip-line. Here are the given details:
- **Horizontal distance (adjacent side to the angle of depression):** 500 feet
- **Length of the zip-line (hypotenuse of the triangle):** 685 feet
We use the trigonometric function cosine (cos) since we have the lengths of the adjacent side and the hypotenuse.
**Cosine Formula:**
\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \]
Substitute the known values:
\[ \cos(\theta) = \frac{500}{685} \]
\[ \cos(\theta) = 0.7299 \]
To find the angle \( \theta \), we take the inverse cosine (arccos):
\[ \theta = \cos^{-1}(0.7299) \]
Using a calculator:
\[ \theta \approx 43.0^\circ \]
Therefore, the angle of depression from Platform A to Platform B is approximately \( 43.0^\circ \) to the nearest tenth.
### Visualization
A right triangle is formed with the following elements:
- The base of the triangle represents the horizontal distance of 500 feet.
- The hypotenuse represents the length of the zip-line which is 685 feet.
- The angle we need to find is the angle of depression from Platform A, looking down to Platform B.
The angle of depression is equal to the angle of elevation of the point on Platform B to Platform A due to parallel horizontal lines and alternate interior angles.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0e956208-9424-4836-8572-da99ead5ba3a%2F5605b2fc-e8f7-4afb-8fd2-921ae395ac9b%2F0ttoya_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**Kara is zip-lining in the rainforest. She is standing at the top of Platform A ready to zip-line to Platform B. If the horizontal distance between the platforms is 500 feet and the length of the zip-line is 685 feet, find the angle of depression from Platform A to Platform B to the nearest tenth.**
### Solution
To find the angle of depression, we can use trigonometric relationships in the right triangle formed by the two platforms and the zip-line. Here are the given details:
- **Horizontal distance (adjacent side to the angle of depression):** 500 feet
- **Length of the zip-line (hypotenuse of the triangle):** 685 feet
We use the trigonometric function cosine (cos) since we have the lengths of the adjacent side and the hypotenuse.
**Cosine Formula:**
\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \]
Substitute the known values:
\[ \cos(\theta) = \frac{500}{685} \]
\[ \cos(\theta) = 0.7299 \]
To find the angle \( \theta \), we take the inverse cosine (arccos):
\[ \theta = \cos^{-1}(0.7299) \]
Using a calculator:
\[ \theta \approx 43.0^\circ \]
Therefore, the angle of depression from Platform A to Platform B is approximately \( 43.0^\circ \) to the nearest tenth.
### Visualization
A right triangle is formed with the following elements:
- The base of the triangle represents the horizontal distance of 500 feet.
- The hypotenuse represents the length of the zip-line which is 685 feet.
- The angle we need to find is the angle of depression from Platform A, looking down to Platform B.
The angle of depression is equal to the angle of elevation of the point on Platform B to Platform A due to parallel horizontal lines and alternate interior angles.
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