Calculating the Distance between Two Ships Based on the Angles of Depression from a Mountain Cliff

Trigonometry is a powerful tool in solving real-world problems, such as calculating distances and heights. This article will walk through an example problem involving angles of depression and two ships positioned on either side of a mountain cliff. We will use basic trigonometric principles to find the distance between the two ships. This method is highly effective and can be applied in various fields, including surveying and navigation.

The Problem

A mountain cliff with a height of 180 meters has an angle of depression of 30° and 60° from the top to two different ships. We need to calculate the distance between these two ships using the principles of trigonometry.

Step-by-Step Solution

Let's denote the height of the cliff as ( h 180 ) meters. We have two angles of depression:

( theta_1 30^circ ) for the first ship ( theta_2 60^circ ) for the second ship

We will use the tangent function to find the horizontal distances from the base of the cliff to each ship.

Distance to the First Ship

For the first ship, the tangent of the angle of depression is given by:

( tan 30^circ frac{h}{d_1} )

Rearranging to solve for ( d_1 ):

( d_1 frac{h}{tan 30^circ} frac{180}{frac{1}{sqrt{3}}} 180 sqrt{3} approx 311.77 , text{m} )

Distance to the Second Ship

For the second ship, the tangent of the angle of depression is given by:

( tan 60^circ frac{h}{d_2} )

Rearranging to solve for ( d_2 ):

( d_2 frac{h}{tan 60^circ} frac{180}{sqrt{3}} approx 103.92 , text{m} )

Calculating the Distance between the Two Ships

The total distance between the two ships is the sum of the distances ( d_1 ) and ( d_2 ).

( text{Distance between the ships} d_1 d_2 180 sqrt{3} frac{180}{sqrt{3}} )

To simplify, we find a common denominator:

( text{Distance} 180 left( sqrt{3} frac{1}{sqrt{3}} right) 180 left( frac{3 1}{sqrt{3}} right) 180 left( frac{4}{sqrt{3}} right) frac{720}{sqrt{3}} approx 416.15 , text{m} )

Hence, the distance between the two ships is approximately 416.15 meters.

Another Example

In another scenario, the height of the mountain cliff above the water surface is 180 meters, and the angles of depression from the top of the cliff to two ships are 30° and 60°. We can calculate the distance between the two ships using the cotangent function:

( text{Distance between A and B} 180 cot 30^circ - cot 60^circ )

Using the values of cotangent:

( cot 30^circ sqrt{3} )

( cot 60^circ frac{1}{sqrt{3}} )

( text{Distance between A and B} 180 sqrt{3} - frac{1}{sqrt{3}} 180 sqrt{3} left( 1 - frac{1}{3sqrt{3}} right) 180 sqrt{3} left( frac{2sqrt{3}}{3} right) frac{415.6921938}{1} )

Hence, the distance between the two ships in this scenario is approximately 415.69 meters.

Conclusion

Trigonometry, especially dealing with angles of depression, is a practical skill for solving real-world problems involving distances and heights. This method can be applied in various fields to achieve accurate results. Whether it is surveying, navigation, or any other application, understanding and utilizing these principles are invaluable.