Two ships are sailing in the sea on the two sides of a lighthouse. The angle of elevation of the top of the lighthouse is observed from the ships are 30 degrees and 45 degrees respectively. If the lighthouse is 100m high, the distance btn the ship is?
It is not possible to answer this, as there are an infinite number of solutions.
Each ship's position is an arc of radius
100ctnTheta (where theta is either 30 or 45). Then if each ship can be anywhere on its arc, the distance between those many possible positions is infinite.
Now if you mean by "on the two sides of a lighthouse" to mean a straight line including the two ships and the lighthouse, then distance between ships is 100(ctnTheta1+ctnTheta2)
To find the distance between the two ships, we can use trigonometry. Let's call the distance between one ship and the lighthouse "x", and the distance between the other ship and the lighthouse "y".
We can set up two equations based on the angles of elevation:
In the first ship's case:
tan(30 degrees) = (height of the lighthouse) / x
In the second ship's case:
tan(45 degrees) = (height of the lighthouse) / y
Using these equations, we can solve for x and y.
For the first equation, we have:
tan(30 degrees) = 100 / x
Simplifying:
√3 / 3 = 100 / x
Cross-multiplying:
√3 * x = 300
Dividing both sides by √3:
x = 300 / √3
For the second equation, we have:
tan(45 degrees) = 100 / y
Simplifying:
1 = 100 / y
Cross-multiplying:
y = 100
So the distance between the two ships is approximately 300 / √3 meters, which is approximately 173.2 meters.