TEAM CHALLENGE It is \( 11: 55 \) a.m., and Emily has climbed even higher. The rope now makes an angle of \( 6^{\circ} \) with the cliff wall. If David is 18 feet away from the base of El Capitan, at what angle should Nathan (who is 185 feet from the base) look up to see Emily?

To determine the angle at which Nathan should look up to see Emily, we first need to identify the height Emily has reached. Since we know the angle of the rope with the cliff wall is \( 6^\circ \) and the distance from the base is the horizontal distance. Let's say, for instance, Emily is \( x \) feet high along the rope. Using trigonometry, particularly the tangent function, we can find Emily’s height using the given angle and David's distance from the base: \[ \tan(6^\circ) = \frac{\text{height}}{18} \] Solving for height gives: \[ \text{height} = 18 \cdot \tan(6^\circ) \approx 18 \cdot 0.1045 \approx 1.88 \text{ ft}. \] Now, Nathan is 185 feet away from the base, and he wants to find the angle \( \theta \) to see Emily who is approximately 1.88 feet up. We can use the tangent function again: \[ \tan(\theta) = \frac{1.88}{185}. \] Thus, \[ \theta = \tan^{-1}\left(\frac{1.88}{185}\right) \approx \tan^{-1}(0.01016) \approx 0.58^\circ. \] Therefore, Nathan should look up at approximately \( 0.58^\circ \) to see Emily. Enjoy the beautiful view of El Capitan!