Huang is standing 14 meters from the base of a kite. The kite string makes a 67° angle with the ground. Find x, the length of the kite string. Round to the nearest hundredth.

Question

(1 point)
Responses

x=69.05 ft.
x=69.05 ft.

x=35.83 ft.
x=35.83 ft.

x=5.47 ft.
x=5.47 ft.

x=15.21 ft.

Answer 1

To find the length of the kite string \( x \), we can use trigonometry. We have a right triangle where:

• The distance from the base of the kite (14 meters) is the adjacent side.
• The angle with the ground is \( 67^\circ \).
• The kite string (length \( x \)) is the hypotenuse.

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

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

Substituting the known values:

\[ \cos(67^\circ) = \frac{14}{x} \]

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

\[ x = \frac{14}{\cos(67^\circ)} \]

Now, we can calculate \( x \). First, we need to find \( \cos(67^\circ) \):

\[ \cos(67^\circ) \approx 0.3907 \]

So now we can substitute this value back into the equation for \( x \):

\[ x \approx \frac{14}{0.3907} \approx 35.83 , \text{meters} \]

Rounding to the nearest hundredth (which is already in the correct form):

\[ x \approx 35.83 , \text{meters} \]

Thus, the correct response is:

x = 35.83 ft.