17. When the angle of elevation to the Sun is 69', a tower casts a shadow on the ground that is 64.3 feet long. sin 65 = 0.9063 sin 69 = 0.9336 cos 69 = 0.3584 cos 65 = 0.4226 tan 69 = 2.6051 tan 65 = 2.1445 Tower Shadow R Use the information in the table to determine the approximate length of the shadow cast by the tower when the angle of elevation to the Sun is 65°. A. 52.9 ft B. 54.5 ft C. 66.2 ft D. 78.1 ft
17. When the angle of elevation to the Sun is 69', a tower casts a shadow on the ground that is 64.3 feet long. sin 65 = 0.9063 sin 69 = 0.9336 cos 69 = 0.3584 cos 65 = 0.4226 tan 69 = 2.6051 tan 65 = 2.1445 Tower Shadow R Use the information in the table to determine the approximate length of the shadow cast by the tower when the angle of elevation to the Sun is 65°. A. 52.9 ft B. 54.5 ft C. 66.2 ft D. 78.1 ft
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
Related questions
Concept explainers
Ratios
A ratio is a comparison between two numbers of the same kind. It represents how many times one number contains another. It also represents how small or large one number is compared to the other.
Trigonometric Ratios
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.
Question
When the
Use the information in the table to determine the approximate length of the shadow cast by the tower when the angle of elevation to the Sun is 65º.
![### Triangles and Trigonometry: Solving for Shadow Lengths
**Example Problem:**
When the angle of elevation to the Sun is 69°, a tower casts a shadow on the ground that is 64.3 feet long.
Given the following trigonometric values:
- \(\sin 69^\circ = 0.9336\)
- \(\cos 69^\circ = 0.3584\)
- \(\tan 69^\circ = 2.6051\)
- \(\sin 65^\circ = 0.9063\)
- \(\cos 65^\circ = 0.4226\)
- \(\tan 65^\circ = 2.1445\)
Consider the labeled right triangle \(\triangle PQR\), where:
- \(P\) represents the top of the tower,
- \(Q\) is the point where the tower meets the ground,
- \(R\) is the point on the ground where the end of the shadow is.
To find the approximate length of the shadow cast by the tower when the angle of elevation to the sun is 65°, follow these steps:
1. **Identify the given length:**
At \(69^\circ\), the shadow length \(QR\) is 64.3 feet.
2. **Apply tangent definitions:**
\[
\tan(69^\circ) = \frac{PQ}{QR}
\]
\[
\tan(65^\circ) = \frac{PQ}{x}
\]
where \(x\) is the unknown shadow length for \(65^\circ\).
3. **Calculate the height of the tower \(PQ\):**
\[
PQ = QR \cdot \tan(69^\circ) = 64.3 \, \text{ft} \cdot 2.6051 = 167.44 \, \text{ft}
\]
4. **Find the unknown shadow length \(x\):**
\[
x = \frac{PQ}{\tan(65^\circ)} = \frac{167.44 \, \text{ft}}{2.1445} \approx 78.1 \, \text{ft}
\]
**Answer Choices:**
- A. 52.9 ft
- B. 54.5 ft
-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc14456c8-c349-46c2-a794-2f6de4965e41%2Fbfe3850c-90dd-4282-b17e-5003ae3aa746%2F8teknsm_processed.png&w=3840&q=75)
Transcribed Image Text:### Triangles and Trigonometry: Solving for Shadow Lengths
**Example Problem:**
When the angle of elevation to the Sun is 69°, a tower casts a shadow on the ground that is 64.3 feet long.
Given the following trigonometric values:
- \(\sin 69^\circ = 0.9336\)
- \(\cos 69^\circ = 0.3584\)
- \(\tan 69^\circ = 2.6051\)
- \(\sin 65^\circ = 0.9063\)
- \(\cos 65^\circ = 0.4226\)
- \(\tan 65^\circ = 2.1445\)
Consider the labeled right triangle \(\triangle PQR\), where:
- \(P\) represents the top of the tower,
- \(Q\) is the point where the tower meets the ground,
- \(R\) is the point on the ground where the end of the shadow is.
To find the approximate length of the shadow cast by the tower when the angle of elevation to the sun is 65°, follow these steps:
1. **Identify the given length:**
At \(69^\circ\), the shadow length \(QR\) is 64.3 feet.
2. **Apply tangent definitions:**
\[
\tan(69^\circ) = \frac{PQ}{QR}
\]
\[
\tan(65^\circ) = \frac{PQ}{x}
\]
where \(x\) is the unknown shadow length for \(65^\circ\).
3. **Calculate the height of the tower \(PQ\):**
\[
PQ = QR \cdot \tan(69^\circ) = 64.3 \, \text{ft} \cdot 2.6051 = 167.44 \, \text{ft}
\]
4. **Find the unknown shadow length \(x\):**
\[
x = \frac{PQ}{\tan(65^\circ)} = \frac{167.44 \, \text{ft}}{2.1445} \approx 78.1 \, \text{ft}
\]
**Answer Choices:**
- A. 52.9 ft
- B. 54.5 ft
-
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, geometry and related others by exploring similar questions and additional content below.Recommended textbooks for you
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
Elementary Geometry for College Students
Geometry
ISBN:
9781285195698
Author:
Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:
Cengage Learning
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
Elementary Geometry for College Students
Geometry
ISBN:
9781285195698
Author:
Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:
Cengage Learning