A man, lying down on top of a cliff 40m high observes the angle of depression of a buoy to be 20°. If he us in line with the buoy, calculate the distance between the buoy and the foot of the cliff (which may be assumed to be vertical) to the nearest metre.
40m and 20
To solve this problem, we can use trigonometry and specifically the tangent function. The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side.
Let's label the unknown distance between the buoy and the foot of the cliff as "x". We can consider this as the adjacent side of the angle of depression.
Given that the angle of depression is 20°, the tangent of this angle is:
tan(20°) = x / 40
To determine the value of x, we can rearrange the equation:
x = 40 * tan(20°)
Now, let's calculate the value of x:
x = 40 * tan(20°)
x ≈ 14.28 meters
Therefore, the distance between the buoy and the foot of the cliff is approximately 14 meters when rounded to the nearest meter.
To find the distance between the buoy and the foot of the cliff, we can use trigonometry.
Let's draw a diagram to visualize the problem:
```
|\
| \ 40m
| \
| \
Buoy | \
| \
|____\
Cliff
```
In the diagram, the observer is lying on top of the cliff, and the buoy is at the same horizontal level as the observer.
We are given that the angle of depression, which is the angle formed between the horizontal line and the line of sight from the observer to the buoy, is 20°.
We can use the tangent function to relate the angle of depression with the distance between the buoy and the foot of the cliff.
tan(angle of depression) = opposite/adjacent
In this case, the opposite side is the vertical height of the cliff (40m), and the adjacent side is the unknown distance we are trying to find.
Let's use the tangent formula to calculate the distance:
tan(20°) = 40m/adjacent
To isolate the variable, we rearrange the formula:
adjacent = 40m / tan(20°)
Using a scientific calculator, we can calculate the value:
adjacent ≈ 40m / 0.36397 ≈ 109.95m
So, the distance between the buoy and the foot of the cliff is approximately 109.95 meters to the nearest meter.