Question

Kyra is standing 50 feet from the kite she is flying. The string of the kite forms an angle of 27°
with the ground. How high in the air is the kite? (1 point)
Responses

98.13 ft.
98.13 ft.

25.48 ft.
25.48 ft.

22.66 ft.
22.66 ft.

32.18 ft.
32.18 ft.

Answer 1

To find the height of the kite above the ground, we can use the sine function from trigonometry. The sine function relates the angle of a right triangle to the ratio of the opposite side (height of the kite) to the hypotenuse (length of the string).

In this case:
- The adjacent side (distance from Kyra to the base of the kite) is 50 feet.
- The angle is 27°.

We can use the following formula:

\[
\text{Height} = \text{hypotenuse} \cdot \sin(\text{angle})
\]

However, we don't know the length of the string (hypotenuse). To find it, we can use the cosine function, since we know the adjacent side.

\[
\cos(27°) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \Rightarrow \text{Hypotenuse} = \frac{\text{Adjacent}}{\cos(27°)}
\]

First, let's calculate the hypotenuse:

\[
\text{Hypotenuse} = \frac{50}{\cos(27°)}
\]

Next, we can find the height using the sine function.

\[
\text{Height} = \text{Hypotenuse} \cdot \sin(27°)
\]

Calculating the cosine and sine values:

1. Calculate \( \cos(27°) \) and \( \sin(27°) \):
- \( \cos(27°) \approx 0.846 \)
- \( \sin(27°) \approx 0.454 \)

Now calculate the hypotenuse:

\[
\text{Hypotenuse} = \frac{50}{0.846} \approx 59.06 \text{ feet}
\]

Now calculate the height:

\[
\text{Height} = 59.06 \cdot 0.454 \approx 26.83 \text{ feet}
\]

This does not seem to match any provided options. However, if we instead use the tangent function for the height directly, since we know the opposite and adjacent sides:

\[
\tan(27°) = \frac{\text{Height}}{50} \Rightarrow \text{Height} = 50 \cdot \tan(27°)
\]

Calculating \( \tan(27°) \):
- \( \tan(27°) \approx 0.510 \)

Now calculate the height:

\[
\text{Height} = 50 \cdot 0.510 \approx 25.50 \text{ feet}
\]

This rounded value is closest to **25.48 ft.**

Thus, the height of the kite is approximately **25.48 ft.**