Task 1263

The task

Piravena must make a trip from $A$ to $B$, then from $B$ to $C$, then from $C$ to $A$. Each of these three parts of the trip is made entirely by bus or entirely by airplane. The cities form a right-angled triangle as shown, with $C$ a distance of 3000 km from $A$ and with $B$ a distance of 3250 km from $A$. To take a bus, it costs Piravena $\$0.15$ per kilometer. To take an airplane, it costs her a $\$100$ booking fee, plus $\$0.10$ per kilometer. [asy] pair A, B, C; C=(0,0); B=(0,1250); A=(3000,0); draw(A--B--C--A); label("A", A, SE); label("B", B, NW); label("C", C, SW); label("3000 km", (A+C)/2, S); label("3250 km", (A+B)/2, NE); draw((0,125)--(125,125)--(125,0)); [/asy] Determine the distance she travels for her complete trip.