Question
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.
x=69.05 ft. x=35.83 ft. x=15.21 ft. x=5.47 ft.
x=69.05 ft. x=35.83 ft. x=15.21 ft. x=5.47 ft.
Answers
GPT-4o mini
To find the length of the kite string (x), we can use trigonometric relationships. Specifically, we will use the tangent function, which relates the opposite side (height of the kite) to the adjacent side (distance from the base of the kite).
Given:
- Distance from the base of the kite (adjacent side) = 14 meters
- Angle of the kite string with the ground = 67°
We can use the tangent function:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Here, \( \theta = 67^\circ \) and the adjacent side is 14 meters.
First, we find the height (h) of the kite:
\[
\tan(67^\circ) = \frac{h}{14}
\]
So,
\[
h = 14 \cdot \tan(67^\circ)
\]
Now, we can calculate h:
Using a calculator to find \( \tan(67^\circ) \approx 2.35585\):
\[
h \approx 14 \cdot 2.35585 \approx 33.0 \text{ meters}
\]
Now, we can find the length of the kite string (x), which is the hypotenuse of the right triangle formed by the height (h) and the distance from the base (14 meters). We use the sine function:
\[
\sin(67^\circ) = \frac{h}{x}
\]
Thus,
\[
x = \frac{h}{\sin(67^\circ)}
\]
Using \( \sin(67^\circ) \approx 0.92050 \):
\[
x = \frac{33.0}{0.92050} \approx 35.83 \text{ meters}
\]
Now, converting to feet (since the options seem to be in feet; 1 meter = 3.28084 feet):
\[
x \approx 35.83 \, \text{m} \cdot 3.28084 \approx 117.38 \, \text{ft}
\]
No options match correctly. However, if we were simply looking for x in meters, the answer is \( x \approx 35.83 \text{ m} \). If we are looking for options in feet, let's confirm it aligns with any listed.
From your options, the closest and appropriate rounded length from the calculations showed would reasonably be \( 35.83 \, \text{ft} \) before conversion.
Thus, the answer is:
\[
x \approx 35.83 \, \text{ft}
\]
Given:
- Distance from the base of the kite (adjacent side) = 14 meters
- Angle of the kite string with the ground = 67°
We can use the tangent function:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
Here, \( \theta = 67^\circ \) and the adjacent side is 14 meters.
First, we find the height (h) of the kite:
\[
\tan(67^\circ) = \frac{h}{14}
\]
So,
\[
h = 14 \cdot \tan(67^\circ)
\]
Now, we can calculate h:
Using a calculator to find \( \tan(67^\circ) \approx 2.35585\):
\[
h \approx 14 \cdot 2.35585 \approx 33.0 \text{ meters}
\]
Now, we can find the length of the kite string (x), which is the hypotenuse of the right triangle formed by the height (h) and the distance from the base (14 meters). We use the sine function:
\[
\sin(67^\circ) = \frac{h}{x}
\]
Thus,
\[
x = \frac{h}{\sin(67^\circ)}
\]
Using \( \sin(67^\circ) \approx 0.92050 \):
\[
x = \frac{33.0}{0.92050} \approx 35.83 \text{ meters}
\]
Now, converting to feet (since the options seem to be in feet; 1 meter = 3.28084 feet):
\[
x \approx 35.83 \, \text{m} \cdot 3.28084 \approx 117.38 \, \text{ft}
\]
No options match correctly. However, if we were simply looking for x in meters, the answer is \( x \approx 35.83 \text{ m} \). If we are looking for options in feet, let's confirm it aligns with any listed.
From your options, the closest and appropriate rounded length from the calculations showed would reasonably be \( 35.83 \, \text{ft} \) before conversion.
Thus, the answer is:
\[
x \approx 35.83 \, \text{ft}
\]