As observed from the top of 60m high light house the angle of dipression of two ships are 30 and 45.Findthe distance between ships.

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Ashish di

To find the distance between the two ships, we can use the concept of trigonometry and right triangles.

First, let's draw a diagram to represent the situation. We have a light house that is 60m high, and two ships that we observe from the top of the light house. Let's label the height of the light house as 'h' (h = 60m), the distance between the two ships as 'd', and the angles of depression (the angles between the line of sight and the horizontal plane) as 30 degrees and 45 degrees.

|\
| \
| \
| \
| \
h=60m | \
| \
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| 30° \ 45°
______________|________\
d

Now, we can use the concept of trigonometry to find the distance between the ships.

For the ship with a 30-degree angle of depression, we can consider the right triangle formed by the line of sight, the horizontal plane, and the distance between the ships. The opposite side of this triangle is the height of the light house (60m), and the adjacent side is half of the distance between the ships (d/2). Therefore, we can use the tangent function to find the value of d/2:

tan(30°) = opposite / adjacent
tan(30°) = 60 / (d/2)
tan(30°) = 120 / d

Similarly, for the ship with a 45-degree angle of depression, we can consider another right triangle formed by the line of sight, the horizontal plane, and the distance between the ships. The opposite side of this triangle is also the height of the light house (60m), and the adjacent side is the full distance between the ships (d). Therefore, we can use the tangent function to find the value of d:

tan(45°) = opposite / adjacent
tan(45°) = 60 / d

Now, we have two equations:
1) tan(30°) = 120 / d
2) tan(45°) = 60 / d

We can solve these equations simultaneously to find the value of 'd', which represents the distance between the two ships.

Once we find the value of 'd', we can double-check our answer using the Pythagorean theorem. We can calculate the height of the right triangle formed by the line of sight, the horizontal plane, and the distance between the ships. If this height matches the given height of the light house (60m), then we can be confident that our answer is correct.