Question

miguel lets out 123 ft. of kite string. the angle of elevation for the kite is 64*. what is the horizontal distance, x, from miguel to the kite?
a. x = 280.58 ft.
b. x = 136.85 ft.
c. x = 110.55 ft.
d. x = 53.92 ft.

Answer 1

To find the horizontal distance \( x \) from Miguel to the kite, we can use the right triangle formed by the height of the kite, the length of the string, and the horizontal distance. In this triangle, the length of the kite string (123 ft) is the hypotenuse, the angle of elevation to the kite is 64°, and \( x \) is the adjacent side.

We can use the cosine function, which relates the adjacent side to the hypotenuse:

\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]

Substituting the known values:

\[
\cos(64^\circ) = \frac{x}{123}
\]

Now, rearranging to solve for \( x \):

\[
x = 123 \cdot \cos(64^\circ)
\]

Now, let's compute \( \cos(64^\circ) \):

Using a calculator, we find that:

\[
\cos(64^\circ) \approx 0.4384
\]

Now, substituting this value back into the equation for \( x \):

\[
x \approx 123 \cdot 0.4384 \approx 53.92 \text{ ft}
\]

Therefore, the horizontal distance \( x \) from Miguel to the kite is:

\[
\text{d. } x = 53.92 \text{ ft.}
\]