**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.

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.

Simplify h/tan(30).

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**.

**Answer:**

**Step-by-step explanation:**

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:

Therefore:

Approximate. So, the height of the lighthouse is approximately:

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