### 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: I need to find the shortest distance from the base camp to Bruce's final position, which is a straight line connecting them.
This is the hypotenuse of a right triangle with legs of $13-6=7$ km (the net northward distance) and $24$ km (the eastward distance).
To find the hypotenuse, I can use the Pythagorean theorem: $a^2+b^2=c^2$, where $a$ and $b$ are the legs and $c$ is the hypotenuse.
Plugging in the given values, I get $(7)^2+(24)^2=c^2$.
Simplifying, I get $49+576=c^2$.
Adding, I get $625=c^2$.
Taking the square root of both sides, I get $c=\sqrt{625}$.
Since $c$ is a positive distance, I can ignore the negative root and take $c=25$.