From the top of a lighthouse 210 feet high, the angle of depression of a boat is twenty seven degrees. Find the distance from the boat to the foot of the light house. The light house was built at sea level
To solve this problem, we can use trigonometry. Let's call the distance from the boat to the foot of the lighthouse "x".
We can set up a right triangle where the height of the lighthouse (210 feet) is the opposite side, "x" is the adjacent side, and the angle of depression (27 degrees) is the angle between the hypotenuse and the adjacent side.
By using the tangent function (tan), we can relate the angle of depression to the sides of the triangle:
tan(angle) = opposite/adjacent
In this case, tan(27 degrees) = 210/x.
To find the value of 'x', we can rearrange the equation:
x = 210 / tan(27 degrees)
Now, let's calculate the value of 'x':
x = 210 / tan(27 degrees)
x ≈ 407.85 feet
Therefore, the distance from the boat to the foot of the lighthouse is approximately 407.85 feet.
d = the distance
A triangle is formed by the lighthouse, the ground and the boat.
The angle at the top of the lighthouse = 90° - the angle of depression
90° - 27°= 63°
tan 63° = d / 210
d = 210 ∙ tan 63° = 210 ∙ 1.9626105055 = 412.148206155 ft