To determine how high the kite is in the air, we can use trigonometry. We can create a right triangle where:
- The length of the string is the hypotenuse (let's call it \( h \)).
- The height of the kite forms the opposite side of the angle.
- The distance from Kyra to the point directly below the kite on the ground forms the adjacent side (50 feet).
Using the sine function, we have:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
In our case:
\[ \sin(27°) = \frac{\text{height of the kite}}{h} \]
Since Kyra is standing 50 feet away from the kite, we can also find the hypotenuse \( h \) using the cosine function:
\[ \cos(27°) = \frac{50 \text{ ft}}{h} \]
Now we can rearrange this to find \( h \):
\[ h = \frac{50}{\cos(27°)} \]
Calculating \( \cos(27°) \):
\[ h = \frac{50}{\cos(27°)} \approx \frac{50}{0.8480} \approx 58.9 \text{ ft} \]
Now, substituting \( h \) back to find the height of the kite:
\[ \sin(27°) = \frac{\text{height}}{58.9} \] \[ \text{height} = 58.9 \times \sin(27°) \] Calculating \( \sin(27°) \):
\[ \text{height} = 58.9 \times 0.4540 \approx 26.68 \text{ ft} \]
Since this is not one of the options given, let's go back and use the direct calculation from the 50 feet distance:
Using the triangle:
\[ \text{height} = 50 \times \tan(27°) \]
Calculating \( \tan(27°) \):
\[ \tan(27°) \approx 0.510 \]
Now:
\[ \text{height} \approx 50 \times 0.510 \approx 25.5 \text{ ft} \]
So, the height of the kite is approximately 25.48 feet, which corresponds to one of the given options.
Therefore, the answer is:
25.48 ft.
1 answer