See image attached for the diagram I used to represent this scenario.
The distance between the ships, at angles 30 and 45, is 200 meters. The distance between the left ship and the lighthouse is x meters.
We can use trigonometric ratios to solve this problem. We can use the tangent ratio to create an equation with the two angles.
Let's take these two equations and solve for x in both of them.
tan(45) = 1, so we can rewrite this equation.
Multiply x to both sides of the equation.
Multiply x + 200 to both sides and divide h by tan(30).
Subtract 200 from both sides of the equation.
Take Equation I and Equation II and set them equal to each other.
Subtract √3 h from both sides of the equation.
Factor h from the left side of the equation.
Divide both sides of the equation by 1 - √3.
Rationalize the denominator by multiplying the numerator and denominator by the conjugate.
Simplify this equation.
The height of the lighthouse is about 273 meters.
Please refer to the attachment.
In the attachment, h is the height of the lighthouse and x is the distance from the lighthouse to Ship A.
Since the angle of depression from the top of the lighthouse to Ship A is 45°, this means that the angle of elevation from Ship A to the top of the lighthouse is 45°.
Likewise, the angle of elevation from Ship B to the top of the lighthouse is 30°.
So, we will form two right triangles: the smaller, 45-45-90 triangle, and the larger 30-60-90 triangle.
Remember that in 45-45-90 triangles, the two legs are congruent.
Therefore, we can write that:
Next, in 30-60-90 triangles, the longer leg is always √3 times the shorter leg.
In our 30-60-90 triangle, the shorter leg is given by:
And the longer leg is given by:
So, the relationship between the shorter leg and longer leg is:
And since we know that h is equivalent to x, we can write:
Now, we just have to solve for h. We can subtract h from both sides:
Factoring out the h yields:
Approximate. So, the height of the lighthouse is approximately: