math concentration question

I was hoping someone could help me understand this question a little more clearly and perhaps put it in different terms.

13.6 kg child diagnosed with a skin infection

order: Ceftriaxone 408mg IV BID

available: 500mg powder vial for reconstitution

instructions: add 1.8mL of diluent solution (0.9% NS); resulting solution contains 250mg/mL

recommended dose: 50-75mg/kg/day in 1-2 divided dose (max: 2g/day)

administration: administer over 30 mins at a final concentration not to exceed 40mg/mL

so the first half of the questions is asking for volume i would withdraw which i calculated as 1.6mL (408mg/250mg X mL)

the next question asks "what is the maximum concentration of fluid you would dilute the ordered dose in?"

I did (only because my professor told us to): 408mg/40mg X mL = 10.2mL

I guess I'm just a little confused on what this question means. I understand that the reconstituted solution of 250mg/mL greatly exceeds the 40mg/mL maximum, but the "maximum" part confuses me. It makes it seem as if I could dilute it less if I wanted to. Why would you reconstitute a medication only to still have the final concentration be too high? For any question like this would you always just divide the amount ordered by the amount that final concentration shouldn't exceed? My professor tried to explain it to us but honestly didn't do a great job at it.

The problem goes on to ask "at what rate would you set the IV pump to properly delivery this medication?"

10.2mL/0.5 hr = 20mL/hr

so even though you withdraw 1.6 mL that's not the amount that actually needs to be infused?? Sorry if this is a long post. We have our med exam next week and I'm just trying to understand this whole maximum concentration thing.

Any help would be much appreciated! Thanks!!