To solve the problem of finding the distance between the ground and the kite, we can use trigonometric principles, specifically the sine function, as the scenario forms a right triangle.
Here are the detailed steps to solve the problem:
1. Visualizing the Problem and Setting Up the Triangle:
- Right Triangle: Imagine a right triangle where the kite string acts as the hypotenuse.
- Angle of Elevation: The angle of elevation formed by the kite string with the ground is given as 46 degrees.
- Hypotenuse: The length of the kite string is 80 feet, this will be our hypotenuse.
2. Identifying the Known and Unknown Values:
- Angle: The angle of elevation (θ) = 46 degrees.
- Hypotenuse (c): The string length = 80 feet.
- Opposite Side (Height or h): This is the distance between the ground and the kite which we need to find.
3. Using Trigonometric Functions:
- In a right triangle, the sine function is defined as:
[tex]\[
\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}
\][/tex]
- In our context:
[tex]\[
\sin(46^\circ) = \frac{\text{height}}{80 \text{ feet}}
\][/tex]
4. Solving for the Height (h):
- Rearrange the sine function to solve for the opposite side (height):
[tex]\[
\text{height} = 80 \text{ feet} \times \sin(46^\circ)
\][/tex]
5. Calculating the Height:
- Using the value of sin(46 degrees):
[tex]\[
\sin(46^\circ) \approx 0.7193
\][/tex]
- Multiply the hypotenuse by the sine of the angle:
[tex]\[
\text{height} = 80 \text{ feet} \times 0.7193 \approx 57.55 \text{ feet}
\][/tex]
6. Rounding the Result:
- To the nearest foot, the height is approximately 58 feet.
### Final Answer:
The distance between the ground and the kite is approximately 58 feet.
This method uses the properties of right triangles and trigonometric functions to determine the unknown height based on the given angle and hypotenuse.
