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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.

please answer with figure.



User Noelle
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2 Answers

0 votes

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
\big{(} \frac{\text{opposite}}{\text{adjacent}} \big{)} to create an equation with the two angles.


  • \displaystyle \text{tan(45)} = (h)/(x)

  • \displaystyle \text{tan(30)} = (h)/(x+200) }

Let's take these two equations and solve for x in both of them.


\textbf{Equation I}


  • \displaystyle \text{tan(45)} = (h)/(x)

tan(45) = 1, so we can rewrite this equation.


  • \displaystyle 1=(h)/(x)

Multiply x to both sides of the equation.


  • \displaystyle x = h


\textbf{Equation II}


  • \displaystyle \text{tan(30)} = (h)/(x+200) }

Multiply x + 200 to both sides and divide h by tan(30).


  • \displaystyle \text{x + 200} = \frac{h}{\text{tan (30)}}

Subtract 200 from both sides of the equation.


  • \displaystyle \text{x} = \frac{h}{\text{tan (30)}} - 200

Simplify h/tan(30).


  • x=√(3)h - 200


\textbf{Equation I = Equation II}

Take Equation I and Equation II and set them equal to each other.


  • h=√(3)h-200

Subtract √3 h from both sides of the equation.


  • h-√(3)h=-200

Factor h from the left side of the equation.


  • h(1-√(3)) =-200

Divide both sides of the equation by 1 - √3.


  • \displaystyle h=(-200)/(1-√(3) )

Rationalize the denominator by multiplying the numerator and denominator by the conjugate.


  • \displaystyle h=(-200)/(1-√(3) ) \big{(} (1+√(3) )/(1+√(3)) \big{)}

  • \displaystyle h=(-200+200√(3) )/(1-3)

  • \displaystyle h =(-200+200√(3) )/(-2)

Simplify this equation.


  • \displaystyle h=100+100√(3)

  • h=273.20508075

The height of the lighthouse is about 273 meters.

The angles of depression of two ships from the top of the light house are 45° and-example-1
User Avnera
by
8.2k points
3 votes

Answer:


h\approx 273.21 \text{ meters}

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:


h=x

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:


\text{Shorter leg}=h

And the longer leg is given by:


\text{Longer leg}=x+200

So, the relationship between the shorter leg and longer leg is:


\sqrt3h = (x+200)

And since we know that h is equivalent to x, we can write:


\sqrt3h=(h+200)

Now, we just have to solve for h. We can subtract h from both sides:


\sqrt3h-h=200

Factoring out the h yields:


h(\sqrt3-1)=200

Therefore:


\displaystyle h=(200)/(\sqrt3-1)

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


h \approx 273.2050 \text{ meters}

The angles of depression of two ships from the top of the light house are 45° and-example-1
User Nglee
by
8.5k points

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