Question
A 10 ft. tent pole has a support rope that extends from the top of the pole to the ground. The rope and the ground form a 30° angle. How long is the rope, rounded to the tenth place?
(1 point)
Responses
20.0 ft.
20.0 ft.
17.3 ft.
17.3 ft.
11.5 ft.
11.5 ft.
3.0 ft.
(1 point)
Responses
20.0 ft.
20.0 ft.
17.3 ft.
17.3 ft.
11.5 ft.
11.5 ft.
3.0 ft.
Answers
GPT-4o mini
To find the length of the rope, which forms a right triangle with the tent pole and the ground, we can use trigonometric functions. Specifically, we can use the sine function which relates the opposite side (the height of the pole) and the hypotenuse (the length of the rope).
Let:
- \( h = 10 \) ft (the height of the tent pole)
- \( \theta = 30^\circ \) (the angle between the rope and the ground)
- \( r \) (the length of the rope)
Using the sine function:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{r}
\]
Rearranging gives:
\[
r = \frac{h}{\sin(\theta)}
\]
Substituting in the values:
\[
\sin(30^\circ) = \frac{1}{2}
\]
Using this in our equation:
\[
r = \frac{10}{\sin(30^\circ)} = \frac{10}{\frac{1}{2}} = 10 \times 2 = 20 \text{ ft}
\]
Thus, the length of the rope is \( 20.0 \) ft.
So the answer is:
**20.0 ft.**
Let:
- \( h = 10 \) ft (the height of the tent pole)
- \( \theta = 30^\circ \) (the angle between the rope and the ground)
- \( r \) (the length of the rope)
Using the sine function:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{r}
\]
Rearranging gives:
\[
r = \frac{h}{\sin(\theta)}
\]
Substituting in the values:
\[
\sin(30^\circ) = \frac{1}{2}
\]
Using this in our equation:
\[
r = \frac{10}{\sin(30^\circ)} = \frac{10}{\frac{1}{2}} = 10 \times 2 = 20 \text{ ft}
\]
Thus, the length of the rope is \( 20.0 \) ft.
So the answer is:
**20.0 ft.**