Question

a 12 - foot pole is supporting a tent and has a rope attached to the top. the rope is pulled straight and the other end is attached to a peg two foot above the ground. the rope and the pole form an angle that measures 40°, as shown below. which expression shows the length of the rope? $\frac{12}{cos40^{circ}}approx15.7$ feet $\frac{12}{cos40^{circ}} - 2approx13.7$ feet $\frac{10}{cos40^{circ}}approx13.1$ feet $\frac{10}{cos40^{circ}}+2approx15.1$ feet

Explanation:

Step1: Determine vertical distance

The vertical distance from the attachment - point of the rope on the pole to the attachment - point on the peg is $12 - 2=10$ feet.

Step2: Use cosine function

In a right - triangle formed by the pole, the ground, and the rope, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here, $\theta = 40^{\circ}$ and the adjacent side to the $40^{\circ}$ angle is the vertical distance of 10 feet, and the hypotenuse is the length of the rope $l$. So, $\cos40^{\circ}=\frac{10}{l}$, and we can solve for $l$ as $l = \frac{10}{\cos40^{\circ}}$.

Answer:

C. $\frac{10}{\cos40^{\circ}}\approx13.1$ feet