A ship is traveling in a circle around the outside rim of an island surveying a good place to land. If it is 6 miles from a lighthouse at the center of the island and 120° from due north, how far has the ship traveled from the most northern tip of the island to the nearest tenth?

12.6 miles
3.1 miles
7.2 miles
20.2 miles

12.6 miles

Step-by-step explanation:

Here the ship is doing a circular path.

We know that the ship is 120° from the north, and we want to find the distance that the ship has traveled so far from the most northern tip of the island.

This distance will be equal to the length of the arc of 120°

You can find this as:

Arc = (angle*3.14/180)*radius

arc = (120*3.14/180)*6 miles  = 12.56 miles

Now we need to round it to the nearest tenth, so we need to round it up, and we get 12.6 miles

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4