1
please answer with figure.

2 Answer

2

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

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

\textbf{Equation I}

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

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

  • \displaystyle 1=\frac{h}{x}

Multiply x to both sides of the equation.

  • \displaystyle x = h

\textbf{Equation II}

  • \displaystyle \text{tan(30)} = \frac{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=\sqrt{3}h - 200  

\textbf{Equation I = Equation II}

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

  • h=\sqrt{3}h-200

Subtract √3 h from both sides of the equation.

  • h-\sqrt{3}h=-200

Factor h from the left side of the equation.

  • h(1-\sqrt{3}) =-200

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

  • \displaystyle h=\frac{-200}{1-\sqrt{3} }

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

  • \displaystyle h=\frac{-200}{1-\sqrt{3} } \big{(} \frac{1+\sqrt{3} }{1+\sqrt{3}} \big{)}
  • \displaystyle h=\frac{-200+200\sqrt{3} }{1-3}
  • \displaystyle h =\frac{-200+200\sqrt{3} }{-2}

Simplify this equation.

  • \displaystyle h=100+100\sqrt{3}
  • h=273.20508075

The height of the lighthouse is about 273 meters.

Edit
avatar
Kayden Bins
15.5k 3 10 26
answered 9 months ago
2

Answer:

h\approx 273.21 \text{ meters}

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:

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=\frac{200}{\sqrt3-1}

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

h \approx 273.2050 \text{ meters}

Edit
avatar
Cynthia Dooley
15.5k 3 10 26
answered 9 months ago