Alright, let's go through the problem step-by-step.
1. Understand the givens:
- Hudson's height is 3 feet.
- The angle of elevation from Hudson's eyes to the kite is 40 degrees.
- The length of the kite string is 125 feet.
2. Identify what you need to find:
- The height of the kite above Hudson.
- The total height of the kite above the ground.
3. Draw a right triangle:
- Hypotenuse: The length of the kite string, which is 125 feet.
- Angle: The angle of elevation, which is 40 degrees.
- Opposite side (x): The height of the kite above Hudson.
4. Use the sine ratio to find the height above Hudson:
- The sine function is defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle.
- [tex]\(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\)[/tex]
5. Set up the equation using the sine function:
- [tex]\(\sin(40^\circ) = \frac{x}{125}\)[/tex]
- To solve for [tex]\(x\)[/tex]:
[tex]\[ x = 125 \times \sin(40^\circ) \][/tex]
6. Calculate [tex]\(x\)[/tex]:
- Using a calculator:
[tex]\[ x \approx 125 \times 0.6428 = 80.35 \text{ feet} \][/tex]
So, the height of the kite above Hudson is approximately 80.35 feet.
7. Add Hudson's height to find the total height above the ground:
- Hudson's height is 3 feet.
- Therefore, the total height of the kite above the ground:
[tex]\[
\text{Height from ground} = \text{Height from Hudson} + \text{Hudson's height}
\][/tex]
[tex]\[
= 80.35 + 3 = 83.35 \text{ feet}
\][/tex]
Thus, the height of the kite is approximately 83.35 feet above the ground.
