Answer:
273 meters
Step-by-step explanation: 
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.
 to create an equation with the two angles. 
Let's take these two equations and solve for x in both of them.
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tan(45) = 1, so we can rewrite this equation.
Multiply x to both sides of the equation.
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Multiply x + 200 to both sides and divide h by tan(30). 
Subtract 200 from both sides of the equation.
Simplify h/tan(30).
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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.