The base of Problem: A 12-foot ladder is leaned up against a wall. the ladder is 9 feet from the base of the wall. Find the measure the ladder's angle of depression. of
The base of Problem: A 12-foot ladder is leaned up against a wall. the ladder is 9 feet from the base of the wall. Find the measure the ladder's angle of depression. of
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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![### Trigonometry Word Problem
**Problem:**
A 12-foot ladder is leaned up against a wall. The base of the ladder is 9 feet from the base of the wall. Find the measure of the ladder's angle of depression.
**Diagrams and Illustrations:**
The diagram provided features a right triangle. Important annotations in the diagram include:
- The ladder, which is indicated by the hypotenuse of the triangle, measures **12 feet**.
- The horizontal distance from the wall to the base of the ladder is marked as **9 feet**.
- The vertical distance from the ground to where the ladder touches the wall is **not labeled** in the picture but can be determined using trigonometric principles or the Pythagorean theorem.
To provide detailed guidance:
1. Identify and label the triangle's sides:
- Hypotenuse (ladder): \( c = 12 \text{ feet} \)
- Adjacent side (base of the ladder to the wall): \( a = 9 \text{ feet} \)
- Opposite side (height up the wall reached by the ladder): \( b \)
2. Apply the trigonometric functions to find the angle of elevation (or depression). Use the cosine function which relates the adjacent side, hypotenuse, and the angle:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{9 \text{ feet}}{12 \text{ feet}}
\]
3. Calculate the angle \(\theta\) by finding the arccosine:
\[
\theta = \arccos\left(\frac{9}{12}\right)
\]
This simplifies to:
\[
\theta = \arccos\left(0.75\right)
\]
4. Now, solve for \(\theta\):
\[
\theta \approx 41.41^\circ
\]
Therefore, the angle of depression of the ladder is approximately **41.41 degrees**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2f3e5dad-3eef-4af4-8432-92621aeb9e2a%2F5811984e-6a3b-4188-8133-937c198b1a18%2F6c5mrg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Trigonometry Word Problem
**Problem:**
A 12-foot ladder is leaned up against a wall. The base of the ladder is 9 feet from the base of the wall. Find the measure of the ladder's angle of depression.
**Diagrams and Illustrations:**
The diagram provided features a right triangle. Important annotations in the diagram include:
- The ladder, which is indicated by the hypotenuse of the triangle, measures **12 feet**.
- The horizontal distance from the wall to the base of the ladder is marked as **9 feet**.
- The vertical distance from the ground to where the ladder touches the wall is **not labeled** in the picture but can be determined using trigonometric principles or the Pythagorean theorem.
To provide detailed guidance:
1. Identify and label the triangle's sides:
- Hypotenuse (ladder): \( c = 12 \text{ feet} \)
- Adjacent side (base of the ladder to the wall): \( a = 9 \text{ feet} \)
- Opposite side (height up the wall reached by the ladder): \( b \)
2. Apply the trigonometric functions to find the angle of elevation (or depression). Use the cosine function which relates the adjacent side, hypotenuse, and the angle:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{9 \text{ feet}}{12 \text{ feet}}
\]
3. Calculate the angle \(\theta\) by finding the arccosine:
\[
\theta = \arccos\left(\frac{9}{12}\right)
\]
This simplifies to:
\[
\theta = \arccos\left(0.75\right)
\]
4. Now, solve for \(\theta\):
\[
\theta \approx 41.41^\circ
\]
Therefore, the angle of depression of the ladder is approximately **41.41 degrees**.
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