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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Question
![The image depicts a right triangle with one angle measuring 60 degrees. The side opposite the 60-degree angle is labeled as \( k \), and the side adjacent to the 60-degree angle is labeled as 24. The problem statement asks to solve for the length of \( k \) and round it to the nearest whole number.
### Problem Statement
**Solve for the length of \( k \). (Round to the nearest whole number)**
\[ k = \boxed{\text{ }} \]
### Diagram Explanation
- **Right Triangle**: The triangle has one right angle (90 degrees).
- **Labelled Angles**: The right angle is denoted, and another angle is given as 60 degrees.
- **Sides**:
- One side adjacent to the 60-degree angle is labeled as 24.
- The side opposite the 60-degree angle is labeled as \( k \).
### Steps to Solve for \( k \)
1. **Identify the Trigonometric Function**: Use the tangent function, which relates the opposite side to the adjacent side in a right triangle.
\[
\tan(60^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{k}{24}
\]
2. **Calculate \( k \)**:
\[
k = 24 \times \tan(60^\circ)
\]
3. **Use the Value of \(\tan(60^\circ)\)**:
\[
\tan(60^\circ) = \sqrt{3} \approx 1.732
\]
\[
k = 24 \times 1.732 \approx 41.568
\]
4. **Round to the Nearest Whole Number**:
\[
k \approx 42
\]
Therefore, the value of \( k \) is approximately 42.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fab545a03-1d39-4874-800c-bed9ffa7318e%2F146603a1-2a67-435e-b9d8-43c76df97a1d%2F07a6go_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image depicts a right triangle with one angle measuring 60 degrees. The side opposite the 60-degree angle is labeled as \( k \), and the side adjacent to the 60-degree angle is labeled as 24. The problem statement asks to solve for the length of \( k \) and round it to the nearest whole number.
### Problem Statement
**Solve for the length of \( k \). (Round to the nearest whole number)**
\[ k = \boxed{\text{ }} \]
### Diagram Explanation
- **Right Triangle**: The triangle has one right angle (90 degrees).
- **Labelled Angles**: The right angle is denoted, and another angle is given as 60 degrees.
- **Sides**:
- One side adjacent to the 60-degree angle is labeled as 24.
- The side opposite the 60-degree angle is labeled as \( k \).
### Steps to Solve for \( k \)
1. **Identify the Trigonometric Function**: Use the tangent function, which relates the opposite side to the adjacent side in a right triangle.
\[
\tan(60^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{k}{24}
\]
2. **Calculate \( k \)**:
\[
k = 24 \times \tan(60^\circ)
\]
3. **Use the Value of \(\tan(60^\circ)\)**:
\[
\tan(60^\circ) = \sqrt{3} \approx 1.732
\]
\[
k = 24 \times 1.732 \approx 41.568
\]
4. **Round to the Nearest Whole Number**:
\[
k \approx 42
\]
Therefore, the value of \( k \) is approximately 42.
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