### 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 want to find the shortest distance between Bruce's base camp and his final location, which is the hypotenuse of a right triangle with legs $13-6=7$ km north and $24$ km east.
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 values, I get $(7)^2+(24)^2=c^2$, which simplifies to $625=c^2$.
Taking the square root of both sides, I get $c=\sqrt{625}=25$.
So the shortest distance between Bruce and his base camp is 25 kilometers.