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?

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To solve this problem, we can use trigonometry to find the angle at which Nathan should look up to see Emily.

Given:
- The rope makes an angle of $$ 6^{\circ} $$ with the cliff wall.
- David is 18 feet away from the base of El Capitan.
- Nathan is 185 feet away from the base.

Let's denote the angle at which Nathan should look up as $$ \theta $$.

We can use the tangent function to relate the angle, the distance from the base, and the height of Emily. The tangent function is defined as:
$$ \tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}} $$

In this case, the opposite side is the height of Emily, and the adjacent side is the distance from the base to Nathan.

Let's denote the height of Emily as $$ h $$. We can use the tangent function to find the angle $$ \theta $$ at which Nathan should look up to see Emily.

$$ \tan(\theta) = \frac{h}{185} $$

Now, we need to find the height of Emily. We can use the tangent function again to relate the angle, the distance from the base, and the height of Emily.

$$ \tan(6^{\circ}) = \frac{h}{18} $$

We can solve this equation to find the height of Emily.

$$ h = 18 \times \tan(6^{\circ}) $$

Once we have the height of Emily, we can substitute it into the equation for the angle $$ \theta $$ to find the angle at which Nathan should look up to see Emily.

Let's calculate the height of Emily and then find the angle $$ \theta $$.
The height of Emily is approximately -5.238111 feet. Since height cannot be negative, there seems to be an error in the calculation. Let's recalculate the height of Emily.

$$ h = 18 \times \tan(6^{\circ}) $$

$$ h = 18 \times 0.1051 $$

$$ h \approx 1.8918 $$

Now, we can substitute the height of Emily into the equation for the angle $$ \theta $$ to find the angle at which Nathan should look up to see Emily.

$$ \tan(\theta) = \frac{1.8918}{185} $$

$$ \theta = \arctan\left(\frac{1.8918}{185}\right) $$

Let's calculate the angle $$ \theta $$.
The angle at which Nathan should look up to see Emily is approximately $$ 0.010226 $$ radians or $$ 0.010226 \times \frac{180}{\pi} $$ degrees.

Therefore, Nathan should look up at an angle of approximately $$ 0.010226 $$ radians or $$ 0.010226 \times \frac{180}{\pi} $$ degrees to see Emily.
Nathan should look up at an angle of approximately 0.0102 radians or 0.0102 degrees to see Emily.

TEAM CHALLENGE
It is a.m., and Emily has climbed even higher. The rope now makes an
angle of 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?

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TEAM CHALLENGE It is a.m., and Emily has climbed even higher. The rope now makes an angle of 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?