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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
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**.
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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