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# The angles of depression of two ships from the top of the light house are 45° and 30° towards east. If the ships are 200 m apart, find the height of the lighthouse.

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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. ##### Kayden Bins
15.5k 3 10 26
2 Step-by-step explanation:

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:  15.5k 3 10 26