There is a damsel in distress at the top of a tower. A ladder is placed 15 feet away from the base of the tower at a 60 degree angle with the tower Damsel Ladder and the ground. What is the length of the ladder to reach the damsel? 09 ofo 15 feet
There is a damsel in distress at the top of a tower. A ladder is placed 15 feet away from the base of the tower at a 60 degree angle with the tower Damsel Ladder and the ground. What is the length of the ladder to reach the damsel? 09 ofo 15 feet
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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Trigonometric ratios give values of trigonometric functions. It always deals with triangles that have one angle measuring 90 degrees. These triangles are right-angled. We take the ratio of sides of these triangles.
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![### Solving the Ladder Problem Using Trigonometry
There is a damsel in distress at the top of a tower. A ladder is placed 15 feet away from the base of the tower at a 60-degree angle with the tower and the ground. What is the length of the ladder to reach the damsel?
#### Breakdown of the Diagram:
The diagram displays a right-angled triangle formed by the tower, the ground, and the ladder. Here are the key components and dimensions provided:
- **Damsel:** Positioned at the top of the tower, requiring rescue.
- **Ladder:** To be used for the rescue. It is placed such that it forms a 60-degree angle with the tower.
- **Base of the Tower to the foot of the Ladder (Adjacent Side):** 15 feet.
- **Angle between the Ladder and the Tower:** 60 degrees.
#### Trigonometric Calculation:
To find the length of the ladder (the hypotenuse in this context), we use the trigonometric function cosine, which is defined for a right-angled triangle as:
\[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]
Where:
- \( \theta = 60^\circ \)
- Adjacent = 15 feet (the distance from the base of the tower to the foot of the ladder)
- Hypotenuse = Ladder length (which we need to find)
Rearranging the formula to solve for the Hypotenuse (L):
\[ \text{Hypotenuse} = \frac{\text{Adjacent}}{\cos(\theta)} \]
Substituting the known values:
\[ \text{L} = \frac{15 \text{ feet}}{\cos(60^\circ)} \]
Knowing that:
\[ \cos(60^\circ) = 0.5 \]
We can calculate:
\[ \text{L} = \frac{15 \text{ feet}}{0.5} = 30 \text{ feet} \]
#### Conclusion:
The length of the ladder required to reach the damsel at the top of the tower is **30 feet**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc476fe49-bd29-4169-8079-6a3850fe1835%2Fb22e3815-213b-443b-804c-cc3d99d2364d%2Fw6kvtb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving the Ladder Problem Using Trigonometry
There is a damsel in distress at the top of a tower. A ladder is placed 15 feet away from the base of the tower at a 60-degree angle with the tower and the ground. What is the length of the ladder to reach the damsel?
#### Breakdown of the Diagram:
The diagram displays a right-angled triangle formed by the tower, the ground, and the ladder. Here are the key components and dimensions provided:
- **Damsel:** Positioned at the top of the tower, requiring rescue.
- **Ladder:** To be used for the rescue. It is placed such that it forms a 60-degree angle with the tower.
- **Base of the Tower to the foot of the Ladder (Adjacent Side):** 15 feet.
- **Angle between the Ladder and the Tower:** 60 degrees.
#### Trigonometric Calculation:
To find the length of the ladder (the hypotenuse in this context), we use the trigonometric function cosine, which is defined for a right-angled triangle as:
\[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]
Where:
- \( \theta = 60^\circ \)
- Adjacent = 15 feet (the distance from the base of the tower to the foot of the ladder)
- Hypotenuse = Ladder length (which we need to find)
Rearranging the formula to solve for the Hypotenuse (L):
\[ \text{Hypotenuse} = \frac{\text{Adjacent}}{\cos(\theta)} \]
Substituting the known values:
\[ \text{L} = \frac{15 \text{ feet}}{\cos(60^\circ)} \]
Knowing that:
\[ \cos(60^\circ) = 0.5 \]
We can calculate:
\[ \text{L} = \frac{15 \text{ feet}}{0.5} = 30 \text{ feet} \]
#### Conclusion:
The length of the ladder required to reach the damsel at the top of the tower is **30 feet**.
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