To find the height of the kite, we can use the sine function from trigonometry. The sine of an angle in a right triangle is given by the formula:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
In this situation:
- The angle \( \theta \) is 27°.
- The distance from Kyra to the point directly below the kite on the ground (the adjacent side) is 50 feet.
- The height of the kite is the opposite side.
The relationship to find the height (h) using the angle and the hypotenuse (the length of the string) can be expressed as:
\[ h = \text{hypotenuse} \times \sin(\theta) \]
To find the hypotenuse, we can use the cosine function, because we know the adjacent side (50 feet):
\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \]
Rearranging this gives us:
\[ \text{hypotenuse} = \frac{\text{adjacent}}{\cos(\theta)} \]
Now calculating the hypotenuse using \( \theta = 27° \):
\[ \text{hypotenuse} = \frac{50}{\cos(27°)} \]
Calculating \( \cos(27°) \) using a calculator:
\[ \cos(27°) \approx 0.848 \]
So:
\[ \text{hypotenuse} \approx \frac{50}{0.848} \approx 59.03 \text{ feet} \]
Next, we can find the height using the sine of 27°:
\[ h = 59.03 \times \sin(27°) \]
Calculating \( \sin(27°) \):
\[ \sin(27°) \approx 0.454 \]
Now calculating the height:
\[ h \approx 59.03 \times 0.454 \approx 26.81 \text{ feet} \]
Since this might not match the options, let’s go directly to the height using the original triangle relationship we first outlined:
The sine relation could also be formulated simply as follows using the given distance from Kyra (50 feet):
\[ h = 50 \times \tan(27°) \]
Calculating \( \tan(27°) \):
\[ \tan(27°) \approx 0.509 \]
Now we substitute:
\[ h \approx 50 \times 0.509 \approx 25.45 \text{ feet} \]
Thus, the closest answer based on the options provided is:
25.48 ft (which matches closely to our calculation).
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