The capital investment cost for a switchgrass$-$fueled ethanol plant with a capacity of $300,000$ gallons per year is $$5\mathrm{.}6$ million. The costcapacity factor for this particular plant technology is $0\mathrm{.}64$ for capacities ranging from $250,000$ gallons per year to $550,000$ gallons per year. What is the estimated capital investment for a similar ethanol plant with a capacity of $500,000$ gallons per year?

The estimated capital investment for a similar ethanol plant with a capacity of $500,000$ gallons per year is $$7780000.($Round to the nearest ten thousands.$)$