Mathematics, 18.12.2020 06:40 joyceandtrey2781

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.

Answers: 3

Show answers

Answers

Answer from: winstonbendariovvygn

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.

The angles of depression of two ships from the top of the light house are 45° and 30° towards east.

Answer from: oudithe3

$h\approx 273.21 \text{ meters}$

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:

$h \approx 273.2050 \text{ meters}$

Answer from: Quest

cbnygjdyj

step-by-step explanation:

Answer from: Quest

tangent

step-by-step explanation:

Other questions on the subject: Mathematics

Mathematics, 21.06.2019 18:50, trevionc0322

Which of the following values cannot be probabilities? 0.08, 5 divided by 3, startroot 2 endroot, negative 0.59, 1, 0, 1.44, 3 divided by 5 select all the values that cannot be probabilities. a. five thirds b. 1.44 c. 1 d. startroot 2 endroot e. three fifths f. 0.08 g. 0 h. negative 0.59

Answers: 2

continue

Mathematics, 21.06.2019 23:00, tylerchitwood211

Ithink i know it but i want to be sure so can you me out ?

Answers: 1

continue

Mathematics, 21.06.2019 23:30, odellbeckham7171

When a valve is open 281 gallons of water go through it in one hour the number of gallons that would go through in 94 hours is

Answers: 1

continue

Mathematics, 22.06.2019 01:00, madelyncross24

Lines f and g are cut by the transversal d. the measure of 1 = 2x + 35 and the measure of 8 = 10x - 85. what value of x will show that lines f and g are parallel? a. 19 b. 26 c. 18 d. 15

Answers: 1

continue

You know the right answer?

The angles of depression of two ships from the top of the light house are 45° and 30° towards east....