Michael M. answered • 10/19/19
Experienced Math and Science Teacher
In Mark M's answer he made one minor inadvertent mistake and he didn't actually explicitly answer either question, but I actually like his basic approach a lot so I am going to use it too.
The first part, how far has the man walked is easy: 4 + 4 + 7 + 6 + 2 = 23 miles. Well, easy to calculate, not so easy to do - almost marathon distance!
For the second part, we'll use the Cartesian coordinate system as a map, where x and y are measured in miles, and, as one would expect, positive x corresponds to east, positive y to north, negative x to west, and negative y to south. We make the origin the man's starting point, so that his x and y coordinates during his travels show where he is relative to his starting point. So he starts at (0,0).
I take it from Mark's answer that while (a, b) is an ordered pair of x and y coordinates, <a, b> is an ordered pair of change in x and y coordinates. We have these changes, in order:
4 miles E: <4, 0>
4 miles SE: <2√2, -2√2>
7 miles S: <0, -7>
6 miles SW: <-3√2, -3√2>
2 miles E: <2, 0>
These are copied and pasted from Mark's answer; thanks Mark!
For those changes involving √2, they arise from moving diagonally. For walking SE, for example, we assume equal amounts S and E. Thus the man's path in these cases is the hypotenuse of an isosceles right triangle. From the Pythagorean theorem, when the hypotenuse has length 1, the legs will each have length 1/√2.
For the second part of the man's walk, he travels 4 miles SE, so the hypotenuse is 4 miles long rather than 1, and thus the legs are 4/√2 = 4√2/√2√2 = 4√2/2 = 2√2 miles long. Thus his x position changes by +2√2 miles, since E is the positive direction for x, and the y position changes by -2√2 miles, since S is the negative direction for y.
For the fourth part of the man's walk we use the same concepts. The magnitude of the changes in x and y is 3√2 rather than 2√2 because he walks 6 miles rather than 4. The signs of the changes are both negative, since he travels SW, which corresponds to moving down and to the left on our map/grid and thus to decreases in x and y.
To get the overall changes in x and y, we add up the changes for the five steps. For x, this sum is 6 - √2 niles. Mark got 6 - 3√2; evidently he missed the +2√2 from the second step. For y, this sum is -7 - 5√2 miles. Thus, the overall change is <6 - √2, -7 - 5√2>. Since he started at (0,0) his final position is (6 - √2, -7 - 5√2). Since the x value is positive and the y value is negative, he ends up somewhat east and south of his starting point.
However, the second part of the question asks how far he would have walked if he walked straight to his final destination, not where that final destination is. The overall changes in x and y give us the legs of a right triangle, and we need the hypotenuse. So we use the Pythagorean theorem:
d = √((6 - √2)2 + (-7 - 5√2)2)
= √(36 - 12√2 + 2 + 49 + 70√2 + 50)
= √(137 + 58√2)
= 14.8 miles
Note that due to the squaring, the signs of the change in x and y don't affect the result, nor should they. For how far he would travel if he went straight to his final destination, it only matters how big his net changes along the x or east-west axis and the y or north-south axis were, and not their directions.
