### 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}

Given: To calculate: Length of AE.

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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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.

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Question

### 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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