from the top of a vertical 186 foot cliff the angle of depression of a boat is 13 degrees(angle of depression is measured downward from horizontal) how far is the boat from the observer at the top of the cliff?

To find the distance from the observer at the top of the cliff to the boat, we can use trigonometry and specifically the tangent function.

Let's start by drawing a diagram to help visualize the situation:


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-----> BOAT ----- x |
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OBSERVER (Top |
of the Cliff) |


In the above diagram, O represents the observer at the top of the cliff, and B represents the boat. The horizontal line represents the surface of the water, and the straight vertical line represents the cliff.

Now, let's use the given information to find the distance (x) from the observer to the boat.

We know that the angle of depression (measured downward from horizontal) is 13 degrees. This means that the angle between the line of sight from the observer to the boat and the horizon is 13 degrees.

Since we have a right triangle formed by the observer, the boat, and a point directly below the observer on the horizontal line, we can use the tangent function to find the value of x.

The tangent of an angle is equal to the length of the side opposite the angle divided by the length of the side adjacent to the angle.

In this case, the side opposite the angle is the vertical line (186 feet) and the side adjacent to the angle is the distance from the observer to the boat (x). Therefore, we can set up the following equation:

tan(13 degrees) = 186 / x

Now, we can solve for x. Rearranging the equation, we have:

x = 186 / tan(13 degrees)

Using a calculator, we find:

x ≈ 186 / 0.224951

x ≈ 826.28 feet

Therefore, the boat is approximately 826.28 feet away from the observer at the top of the cliff.

To find the distance, we can use the tangent function, which relates the angle of depression to the opposite and adjacent sides of a right triangle.

Let's call the distance between the boat and the observer "d".

We have an opposite side, which is the height of the cliff, and an adjacent side, which is the distance between the boat and the observer.

With the given information, we can set up the equation:

tangent(13 degrees) = opposite/adjacent
tan(13°) = 186/d

Now we can solve for "d" by rearranging the equation:

d = 186 / tan(13°)

Calculating this value:

d ≈ 663.408

Therefore, the boat is approximately 663.408 feet away from the observer at the top of the cliff.

So you need the length of the hypotenuse

sin 13° = 186/x
x = 186/sin13° = 826.8 ft