Help please !! very much needed !!

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

azamat

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.

<h2> $\textbf{Equation I}$ </h2>

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

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

$\displaystyle 1=\frac{h}{x}$

Multiply x to both sides of the equation.

$\displaystyle x = h$

<h2> $\textbf{Equation II}$ </h2>

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

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).

$x=\sqrt{3}h - 200$

<h2> $\textbf{Equation I = Equation II}$ </h2>

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.

5 0

3 years ago

Read 2 more answers

A rectangular vegetable garden will have a width that is feet less than the length, and an area of square feet. If x represents

ivanzaharov [21]

8(8-2)=48
8(6)=48
therefore x=8

5 0

2 years ago

Adem can eat 3/8 of a pizza in 1/16 of a minute. Compute the unit rate by setting up a complex fraction and simplifying

Alchen [17]

Answer:

Can eat 6 pizzas per minute or 1 pizza in 10 seconds. "Wouldn't want to be around Adem trying to swallow that many so quickly."

Step-by-step explanation:

(3/8)/(1/16) = x/1

48/8 = x

x = 6 pizzas per minute

4 0

3 years ago

Solve the equation for y <br> -2x+y=11

boyakko [2]

Hi Dezariaspencer

All we have to do is simply add 2x to both sides and we have our answer.

Answer: y = 11 + 2x.

Thank you for asking your question, have a good day.

3 0

4 years ago

Read 2 more answers

Please help me it takes so long to do this please

leonid [27]

Answer:

x = 117°

Step-by-step explanation:

The angle inside the triangle and 101° are adjacent and supplementary, thus

angle inside triangle = 180° - 101° = 79°

The exterior angle of a triangle is equal to the sum of the 2 opposite interior angles.

x is an exterior angle of the triangle , thus

x = 38° + 79° = 117°

8 0

3 years ago

Read 2 more answers