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.

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To find the distance from the boat to the foot of the lighthouse, we can use trigonometry and the given angle of depression.

First, let's define the variables:
- h = height of the lighthouse (210 feet)
- θ = angle of depression (27 degrees)
- d = distance from the boat to the foot of the lighthouse (unknown)

Now, let's visualize the situation. Imagine a right triangle formed by the lighthouse, the boat, and the direct line between them. The height of the lighthouse is the opposite side of the angle of depression, and the distance we are trying to find is the adjacent side.

Using the tangent function, we can relate the angle of depression and the sides of the triangle:

tan(θ) = opposite/adjacent
tan(27°) = h/d

To solve for d, let's rearrange the equation:

d = h/tan(θ)
d = 210 feet / tan(27°)

Using a calculator, we can find the tangent of 27 degrees:

tan(27°) ≈ 0.5095

Now, let's substitute the values and calculate:

d = 210 feet / 0.5095
d ≈ 411.3 feet

Therefore, the distance from the boat to the foot of the lighthouse is approximately 411.3 feet.

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