From the top of a lighthouse 210 feet high, the angle of depression to a boat is 27. Find the distance from the boat to the food of the lighthouse. The lighthouse was built at sea level.
Help me!
height/distance=tan27
distance= height/tan27
Who would ever build a lighthouse at sea level? You build bouys at sea level.
Sure, I'd be happy to help you with that! To find the distance from the boat to the foot of the lighthouse, we can use trigonometry and specifically, the tangent function.
First, let's visualize the situation. We have a lighthouse that is 210 feet high. From the top of the lighthouse, an observer looks down to the boat and measures an angle of depression of 27 degrees. The goal is to find the horizontal distance from the boat to the foot of the lighthouse.
Now, let's break down the problem. We have a right triangle formed by the observer at the top of the lighthouse, the foot of the lighthouse, and the boat.
The height of the lighthouse (opposite side) is 210 feet, and we want to find the distance from the boat to the foot of the lighthouse (adjacent side). The angle of depression (angle between the hypotenuse and the adjacent side) is 27 degrees.
We can use the tangent function, which is defined as the ratio of the opposite side to the adjacent side in a right triangle. In this case, tan(27°) = opposite / adjacent, or tan(27°) = 210 / adjacent.
To isolate the adjacent side, we can rearrange the equation: adjacent = 210 / tan(27°).
Now, we just need to plug in the values and calculate the solution.
Using a calculator, the tangent of 27 degrees is approximately 0.5095. Therefore, adjacent = 210 / 0.5095 ≈ 412.7 feet.
So, the distance from the boat to the foot of the lighthouse is approximately 412.7 feet.
I hope that helps! Let me know if you have any further questions.