From the top of a lighthouse 210 feet high, the angle if depression to a boat is 27 degress. Find the distance from the boat to the food of the lighthouse. The lighthouse was built at sea level.

I solved this one under a previous post.

To find the distance from the boat to the foot of the lighthouse, we can use trigonometric concepts.

Let's define the following:

h = height of the lighthouse (in this case, h = 210 feet)
θ = angle of depression (in this case, θ = 27 degrees)

We need to find the distance from the boat to the foot of the lighthouse (let's call it d).

The angle of depression is the angle formed by a horizontal line, from the observer (top of the lighthouse), to the line of sight towards the object (the boat), measured downwards. Therefore, we can conclude that the angle between the horizontal line and the line connecting the top of the lighthouse to the foot of the lighthouse is (90 - θ) degrees.

Now, we can use trigonometric ratios, specifically the tangent function, to find the distance d.

Tangent is defined as the ratio of the opposite side to the adjacent side in a right triangle. In this case, the opposite side is the height of the lighthouse (h) and the adjacent side is the distance we want to find (d). So we have:

tan(90 - θ) = h / d

Substituting the given values:

tan(90 - 27) = 210 / d

tan(63) = 210 / d

To solve for d, we can rearrange the equation as follows:

d = 210 / tan(63)

Using a scientific calculator, we can find the value of tan(63) to be approximately 2.2777.

Substituting this value into the equation, we have:

d = 210 / 2.2777

Finally, we can calculate the value of d:

d ≈ 92.15 feet

Therefore, the distance from the boat to the foot of the lighthouse is approximately 92.15 feet.