Question

answer fast please will give brainliest

A ship is sailing due north. The captain sees a lighthouse 12km away on a bearing of 036°. Later, the captain observes that the bearing of the lighthouse is 125°. How far did the ship travel between these two observations?

Answer 1

Answer:

Approximately 14.65 kilometers.

Step-by-step explanation:

Please refer to the drawing.

So, to find our distance, we essentially just need to find the value of x.

To do so, we can use the Law of Sines.

Note that ∠a corresponds to the 12km, and ∠c corresponds to x.

So, let's find each of the angles first.

Note that from our ending point, the bearing to the lighthouse is 125°.

This means, if you refer to the drawing:

$a+b=90$

This is because they are complementary.

We also know that:

$90+b=125$

From our bearing.

So, let's solve for b. Subtract 90 from both sides:

$b=35$

Substitute this back into our first equation to acquire:

$a+35=90$

Subtract 35 from both sides:

$a=55\textdegree$

Now, we can use the fact that the interior angles of a triangle sum to 180 to find c. We already know that one angle is 36 and the other is 55. So:

$36+55+c=180$

Add:

$91+c=180$

Subtract 91 from both sides:

$c=89\textdegree$

Now, we can use the Law of Sines, which states:

$\displaystyle \frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c}$

It doesn't really matter which ratio we use as long as we align the sides and angles correctly.

Recall that ∠a correspond to 12km while ∠c corresponds to x.

So, let's substitute 12 for a, 55 for A, x for b, and 89 for B. This yields:

$\displaystyle \frac{\sin(55)}{12}=\frac{\sin(89)}{x}$

So, to find the distance traveled, we just have to find x.

Cross-multiply:

$12\sin(89)=x\sin(55)$

Divide both sides by sin(55). This yields:

$\displaystyle x=\frac{12\sin(89)}{\sin(55)}$

Divide. Use a calculator. Make sure you're in Degrees mode. This gives us:

$x\approx14.65$

So, the ship approximately traveled 14.65 kilometers from Point A to Point B.

And we're done!