### Instruction: From his base camp at the equator, Bruce hiked on flat terrain $13$ kilometers due north, then $24$ km due east, and finally $6$ km due south. Using a direct route, how many kilometers would Bruce be from base camp?  (Assume the Earth is flat for this problem.).### Response: The problem asks for the distance from the base camp to the final location of Bruce, which is a direct route.
This suggests using the Pythagorean theorem, since the base camp, the final location, and the eastward endpoint of the hike form a right triangle.
To apply the Pythagorean theorem, we need to know the lengths of the legs of the right triangle.
One leg is the eastward distance of $24$ km.
The other leg is the net northward distance of Bruce, which is $13 - 6 = 7$ km, since he went north $13$ km and then south $6$ km.
The Pythagorean theorem says that the square of the hypotenuse, which is the distance we want, is equal to the sum of the squares of the legs.
So we have:

\[d^2 = 24^2 + 7^2\]
Simplifying, we get:

\[d^2 = 576 + 49\]

\[d^2 = 625\]
Taking the square root of both sides, we get:

\[d = \sqrt{625}\]