Due to weather, a barge captain decides to reach her destination in two legs: one due north and one due west. On a direct route, her destination is about 1{,}8301,8301, comma, 830 miles \text{(mi)}(mi)start text, left parenthesis, m, i, right parenthesis, end text away; see the figure above. If after traveling 605 \text{ mi}605 mi605, start text, space, m, i, end text due north the captain determines it is time to head due west, how many more miles are left in the trip? (Round the answer to the nearest mile.)

Notice that the 830 miles direct trip would be the hypotenuse of a right angle triangle. The captain goes due North for 605 mi, and then needs to turn west.

Please see the attache image to guide you.

So the question is what is the length of the third side (x) of that right angle triangle which accounts for the distance that has to be sailed due west. So we use the Pythagorean theorem:

[tex]hyp^2=leg_1^2+leg_2^2\\830^2=605^2+x^2\\x^2=830^2-605^2\\x=\sqrt{830^2-605^2} \\x=568.22\,\,mi[/tex]