Question

HELP! LOTS OF POINTS!

Answer 1

D(last option)

Explanation:

This question might be a bit confusing since a lot of questions like this one require actually visualizing the question. The man in the lighthouse isn't trying to throw the rope directly in front of him to the birds, but rather to the boat.

The angle of depression here is the angle formed from the rope and the man's line of sight. Therefore, the angle of depression is congruent to the angle's alternate interior counterpart, which is the angle of elevation formed from the boat to the water.

From there, we solve like an angle of elevation question. We have the rope length, which is the hypothenuse, but we need to find the length of the side adjacent to the angle of elevation. So, we use cosine.

We just need to plug in the values here. The measure of the angle goes right next to the trig function we are using, and in cosine, the measure of the adjacent angle goes over the measure of the hypothenuse.

Therefore, the answer is D, or the last option.

Answer 2

`\textsf{D)} \quad \cos \left(24^(\circ)\right)=(x)/(45)`

Explanation:

The angle of depression is the angle between the horizontal line of sight of the man in the lighthouse, and the direct line of sight to the boat below him.

To find how far from land the boat is, we can model the scenario as a right triangle.

The angle of depression (24°) is the angle formed by the hypotenuse and the horizontal leg of the right triangle.

If the distance to throw a rope from the top of the lighthouse to the boat is 45 feet, then the hypotenuse of the triangle is 45 feet.

The distance between the boat and the land is the side adjacent to the angle, so we can find this by using the cosine trigonometric ratio.

`\boxed{\begin{minipage}{9 cm}\underline{Cosine trigonometric ratio} \\\\$\sf \cos(\theta)=(A)/(H)$\\\\where:\\ \phantom{ww}$\bullet$ $\theta$ is the angle. \\ \phantom{ww}$\bullet$ $\sf A$ is the side adjacent the angle. \\\phantom{ww}$\bullet$ $\sf H$ is the hypotenuse (the side opposite the right angle). \\\end{minipage}}`

Given:

• θ = 24°
• A = x
• H = 45

Substitute these values into the ratio to create the required equation:

`\large\boxed{\cos \left(24^(\circ)\right)=(x)/(45)}`