jmc

Since $△ABC$ is a right-angled triangle, then we may use the Pythagorean Theorem to find $BC$. Thus, $A{B}^2=B{C}^2+C{A}^2$, and so $$\begin{array}{rl}B{C}^2& =A{B}^2-C{A}^2\\ & =3250^2-3000^2\\ & =250^2(13^2-12^2)\\ & =250^2(5^2)\\ & =1250^2.\end{array}$$therefore $BC=1250$ km (since $BC>0$).

To fly from $A$ to $B$, the cost is 3250\times0.10+100=\425$.Tobusfrom$A$to$B$,thecostis$3250\times0.15=\$487.50$. Since Piravena chooses the least expensive way to travel, she will fly from $A$ to $B$.

To fly from $B$ to $C$, the cost is 1250\times0.10+100=\225$.Tobusfrom$B$to$C$,thecostis$1250\times0.15=\$187.50$. Since Piravena chooses the least expensive way to travel, she will bus from $B$ to $C$.

To fly from $C$ to $A$, the cost is 3000\times0.10+100=\400$.Tobusfrom$C$to$A$,thecostis$3000\times0.15=\$450$. Since Piravena chooses the least expensive way to travel, she will fly from $C$ to $A$.

The total cost of the trip would be \425+\187.50+\400=\boxed{\1012.50}.