From the top of a cliff 150m high, the angles of depression of two boats are 60 and 30. Find the distance between the boats if they are on the opposite sides of the cliff.

To find the distance between the boats, we can use trigonometry and the concept of angles of depression.

Let's draw a diagram to visualize the situation:

C
|\
| \
| \ 150m
| \
| \
A | \ B
| \
----------------|-------\
30° 60°

In the diagram, A represents one boat, B represents the other boat, and C represents the top of the cliff.

Given that the angle of depression from the top of the cliff to boat A is 60° and to boat B is 30°, we can use trigonometry to find the distances AC and BC.

Using the tangent function:

tan(θ) = opposite/adjacent

For boat A, the opposite side is the height of the cliff (AC = 150m) and the adjacent side is the distance we want to find (AB).

tan(60°) = AC/AB

tan(60°) = 150/AB

Solving for AB, we get AB = 150/tan(60°)

Similarly, for boat B:

tan(30°) = BC/AB

tan(30°) = 150/AB

Solving for AB, we get AB = 150/tan(30°)

Now we have two equations with AB on the right side, so we can equate them:

150/tan(60°) = 150/tan(30°)

Now we can find the value of AB by solving the above equation:

AB = (150/tan(60°)) * tan(30°)

Using a scientific calculator, we find that tan(60°) ≈ 1.732 and tan(30°) ≈ 0.577.

AB = (150/1.732) * 0.577

Simplifying the expression, we get:

AB ≈ 86.60 meters

Therefore, the distance between the two boats on the opposite sides of the cliff is approximately 86.60 meters.

To solve this problem, we can use trigonometric ratios. Let's label the distance between the boats as 'x'.

First, let's consider the boat with an angle of depression of 60 degrees. From the top of the cliff, the vertical distance to this boat is 150m. We can use the trigonometric ratio for tangent:

tan(60) = opposite/adjacent
tan(60) = 150/x
√3 = 150/x
x = 150/√3

Next, let's consider the boat with an angle of depression of 30 degrees. From the top of the cliff, the vertical distance to this boat is also 150m. Again, using the trigonometric ratio for tangent:

tan(30) = opposite/adjacent
tan(30) = 150/(x+150)
1/√3 = 150/(x+150)
x+150 = 150/√3
x+150 = 150√3/3
x+150 = 50√3
x = 50√3 - 150

So, the distance between the boats, when they are on opposite sides of the cliff, is x = 150/√3 or approximately x = 50√3 - 150 (in the same units as the given height, which is 150m).

tan60 = Y/X1 = 150/X1.

X1 = 150/tan60 = 86.6 m. = Dist. from
1st boat to bottom of cliff.

tan30 = Y/X2 = 150/X2.
X2 = 150/tan30 = 259.8 m. = Dist. from 2nd boat to bottom of cliff.

X = X1 + X2 = 86.6 + 259.8 = 346.4 m. = Dist. between the two boats.