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.

Draw a rectangle with the long sides

hor and the short sides(210ft) ver.
Draw a diagonal from the upper left hand corner to the lower rt corner.

The angle between the diagonal and the
top hor side is the angle of depression.

tan27 = Y/X,
tan27 = 210/X,
X = 210 / tan27 = 412ft = dist. from
boat to foot of lighthouse.

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To find the distance from the boat to the foot of the lighthouse, we can use trigonometric ratios and right triangle properties.

Let's label the distance we want to find as 'x'. We know that the height of the lighthouse is 210 feet and the angle of depression to the boat is 27 degrees. We can visualize this as a right triangle, with the height of the lighthouse as the perpendicular side, the unknown distance 'x' as the base, and the line of sight from the top of the lighthouse to the boat as the hypotenuse.

Now, let's use the trigonometric ratio for tangent. The tangent of an angle is equal to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

In this case, the tangent of 27 degrees is equal to the height of the lighthouse divided by the distance 'x':

tan(27 degrees) = 210 feet / x

To find 'x', we can rearrange the equation:

x = 210 feet / tan(27 degrees)

Now, we can plug in the values and calculate:

x ≈ 210 feet / 0.5095

x ≈ 411.20 feet

Therefore, the distance from the boat to the foot of the lighthouse is approximately 411.20 feet.