From the top of a lighthouse 120 m above the sea, the angle of
depression of a boat is 15°. How far is the boat from the lighthouse?
Provide an Illustration.
Draw the diagram. It should be clear that
120/x = tan15°
To find the distance between the boat and the lighthouse, we can use trigonometric ratios. In this case, we will use the tangent function.
Let's start by drawing a diagram to better visualize the situation. We have a lighthouse on top of a cliff 120 m above the sea level, and there's a boat somewhere on the sea below. The angle of depression, which is the angle between the line of sight from the top of the lighthouse to the boat and the horizontal line, is 15°.
```
|
| /
| /
Lighthouse /
| /
------------|/--------- Sea Level
|
|
Boat
```
Now, let's denote the distance between the lighthouse and the boat as 'x'. We can use the tangent function to relate the angle of depression and the distance.
The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the lighthouse (120 m) and the adjacent side is the distance we're trying to find (x).
So, we have the equation:
tan(15°) = 120 / x
To solve for x, we rearrange the equation:
x = 120 / tan(15°)
Using a calculator, we can find the value:
x ≈ 410.17 m
Therefore, the boat is approximately 410.17 meters away from the lighthouse.
Note: Make sure that your calculator or device is set to degrees mode when using trigonometric functions as the angle measurement is in degrees.