from the top of a light house 40 m above the water the angle of depression of a small boat is 60 degree .find how far the boat is from the base of the light house
i want the answer please full detailed
To find the distance between the boat and the base of the lighthouse, we can use trigonometry. Specifically, we can use the tangent function.
The angle of depression is the angle between the line of sight from the top of the lighthouse to the boat and the horizontal line. In this case, the angle of depression is 60 degrees.
Let's label the base of the lighthouse as point A, the top of the lighthouse as point B, and the boat as point C.
We can define a right triangle with sides AB, BC, and AC. The height of the lighthouse (AB) is given as 40 m, and the angle of depression (angle BAC) is 60 degrees.
The tangent of an angle is equal to the length of the opposite side divided by the length of the adjacent side. In this case, the length of the opposite side is AB (40 m) and the length of the adjacent side is BC (the distance we need to find).
So, we have the equation:
tan(60 degrees) = 40 m / BC
Now we can solve for BC:
BC = 40 m / tan(60 degrees)
Using a calculator, we find that tan(60 degrees) is approximately 1.732. So, the equation becomes:
BC = 40 m / 1.732
Calculating this, we find that BC is approximately 23.1 meters.
Therefore, the boat is approximately 23.1 meters away from the base of the lighthouse.
BC/AB=cos
20/40=cos
1/2=cos
Then cos=60