One adult traveling with an eleven-year-old can get from Manhattan to JFK for $24.50, utilizing the LIRR ($7.00 for a "City Ticket Peak" ticket, plus $1.00 for a "Family Fare" child ticket; and $16.50 for two passengers on the JFK AirTrain). I would allow about one hour for the trip. Of course, if you need to take a taxi from your starting point to the train station, add in that time and cost.

The LIRR now serves both Penn Station and Grand Central Terminal. LIRR trains depart Grand Central from the northern part of the terminal; easiest access is the entrance on Madison Avenue and 47th Street. More information here:

https://new.mta.info/agency/long-isl...-madison-guide

("Everything you need to know about Grand Central Madison")

A yellow cab from Midtown to JFK during the afternoon rush hour would cost ~ $100, including surcharges, tolls, and the customary 15-20% tip. Travel time is unpredictable, depending on rush-hour traffic conditions -- including how long it would take to get from your starting point to the mouth of the Queens-Midtown Tunnel.