A boy is flying a kite. His kite string is extended 50 meters long. The boy is 1.7 meters tall, and the angle of elevation of the kite to the boy's hands is 12°. How high is the kite flying above the ground? 50m 12° 1.7 m
A boy is flying a kite. His kite string is extended 50 meters long. The boy is 1.7 meters tall, and the angle of elevation of the kite to the boy's hands is 12°. How high is the kite flying above the ground? 50m 12° 1.7 m
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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![### Geometry Problem: Calculating the Height of a Kite
**Problem Statement:**
A boy is flying a kite. His kite string is extended 50 meters long. The boy is 1.7 meters tall, and the angle of elevation of the kite to the boy's hands is 12°. How high is the kite flying above the ground?
**Diagram Explanation:**
The diagram illustrates a boy flying a kite. There are several key measurements and angles depicted:
1. **Kite String Length (Hypotenuse)**: The kite string, which is the hypotenuse of the right triangle, is 50 meters long.
2. **Angle of Elevation**: The angle between the string and the horizontal ground is 12°.
3. **Boy's Height**: The height of the boy from the ground to his hands is 1.7 meters.
4. **Vertical Height (x)**: The distance from the boy’s hands to the kite, vertically aligned, is represented as \( x \).
The goal is to determine the total height of the kite above the ground.
**Step-by-Step Solution:**
1. **Establish the Right Triangle**:
- The hypotenuse is 50 meters.
- The angle of elevation is 12°.
2. **Calculate the Vertical Component**:
Using the sine function for the angle of elevation:
\[
\sin(12^\circ) = \frac{\text{Opposite Side (Vertical Component)}}{\text{Hypotenuse (Kite String Length)}}
\]
Let \( h \) be the vertical component from the boy's hands to the kite:
\[
\sin(12^\circ) = \frac{h}{50}
\]
Therefore:
\[
h = 50 \times \sin(12^\circ)
\]
Using a calculator:
\[
h \approx 50 \times 0.2079 \approx 10.395 \text{ meters}
\]
3. **Total Height Above Ground**:
Add the boy's height to the vertical component, \( x \):
\[
\text{Total Height} = h + \text{Boy's Height}
\]
\[
\text{Total Height} \approx 10.395 + 1.7 \approx 12.095 \text{ meters}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc476fe49-bd29-4169-8079-6a3850fe1835%2F4fc80bcf-0eab-435e-b415-4a86b3f5f2ac%2Fg2j5goj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Geometry Problem: Calculating the Height of a Kite
**Problem Statement:**
A boy is flying a kite. His kite string is extended 50 meters long. The boy is 1.7 meters tall, and the angle of elevation of the kite to the boy's hands is 12°. How high is the kite flying above the ground?
**Diagram Explanation:**
The diagram illustrates a boy flying a kite. There are several key measurements and angles depicted:
1. **Kite String Length (Hypotenuse)**: The kite string, which is the hypotenuse of the right triangle, is 50 meters long.
2. **Angle of Elevation**: The angle between the string and the horizontal ground is 12°.
3. **Boy's Height**: The height of the boy from the ground to his hands is 1.7 meters.
4. **Vertical Height (x)**: The distance from the boy’s hands to the kite, vertically aligned, is represented as \( x \).
The goal is to determine the total height of the kite above the ground.
**Step-by-Step Solution:**
1. **Establish the Right Triangle**:
- The hypotenuse is 50 meters.
- The angle of elevation is 12°.
2. **Calculate the Vertical Component**:
Using the sine function for the angle of elevation:
\[
\sin(12^\circ) = \frac{\text{Opposite Side (Vertical Component)}}{\text{Hypotenuse (Kite String Length)}}
\]
Let \( h \) be the vertical component from the boy's hands to the kite:
\[
\sin(12^\circ) = \frac{h}{50}
\]
Therefore:
\[
h = 50 \times \sin(12^\circ)
\]
Using a calculator:
\[
h \approx 50 \times 0.2079 \approx 10.395 \text{ meters}
\]
3. **Total Height Above Ground**:
Add the boy's height to the vertical component, \( x \):
\[
\text{Total Height} = h + \text{Boy's Height}
\]
\[
\text{Total Height} \approx 10.395 + 1.7 \approx 12.095 \text{ meters}
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