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.
Check 4-18-11,1:42pm post for solution.
To find the distance from the boat to the foot of the lighthouse, we can use trigonometry and the concept of angle of depression.
The angle of depression is the angle formed by a horizontal line and the line of sight from an observer to an object below the horizontal line. In this case, the angle of depression is 27 degrees.
Let's consider the triangle formed by the top of the lighthouse, the foot of the lighthouse, and the boat. The height of the lighthouse is given as 210 feet, and we need to find the distance from the boat to the foot of the lighthouse.
Using the concept of tangent, we know that the tangent of an angle of a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. In this case, the opposite side is the height of the lighthouse (210 feet), and the adjacent side is the distance from the boat to the foot of the lighthouse (which we need to find).
So, we can set up the equation:
tan(27 degrees) = opposite/adjacent
Substituting the values we know:
tan(27 degrees) = 210 feet / adjacent
To solve for the adjacent side (distance from the boat to the foot of the lighthouse), we can multiply both sides of the equation by the adjacent side:
adjacent * tan(27 degrees) = 210 feet
Now, divide both sides of the equation by tan(27 degrees) to isolate the adjacent side:
adjacent = 210 feet / tan(27 degrees)
Using a scientific calculator, calculate the tangent of 27 degrees, which is approximately 0.5095. Now, insert this value into the equation:
adjacent = 210 feet / 0.5095
Calculating this expression, we find that the distance from the boat to the foot of the lighthouse is approximately 412.14 feet.
So, the distance from the boat to the foot of the lighthouse is approximately 412.14 feet.