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A century-old idea to extend the subway from Crown Heights to Marine Park is picking up steam.

The Metropolitan Transportation Authority set aside $5 million to study laying tracks along Utica Avenue last year, raising eyebrows as to why the cash-strapped authority was funding investigations for future construction when it couldn’t get the city and state to agree on how to pay for current projects. But the transit agency is actually hoping the Feds will pick up the tab for the look-see, state documents reveal.

The state Department of Transportation is asking Uncle Sam for $4.6–$7.5 million on behalf of the subway authority to consider stretching the 2 and 5 trains from Eastern Parkway to Avenue U — mostly along Utica Avenue — through a subway-starved section of Brooklyn, according to a recently published list of proposed ventures through 2020 called the Statewide Transportation Improvement Program.

And straphangers say the plan would be a boon for Marine Parkers.

“It would be, because you can commute more quickly,” said Michelle, who lives on E. 54th Street and Avenue U but declined to give a last name. “Instead of relying solely on the bus, you can choose between bus and subway. We all know the subway goes more quickly.”

The study would be a first by the authority — even though the notion has been around since the early 1900s — a spokesman from the Metropolitan Transportation Authority said.

But the whole notion is a colossal waste of time and money, because the authority likely can’t afford to build the extension, construction would be disruptive, and the line would duplicate the recently rolled-out B46 Select Bus Service, the local community board’s district manager said. Instead, it should use the cash to better existing service rather than duplicating the route.

“This will never become a reality because it will take billions upon billions, the community will probably be put into disarray — you will have to displace businesses, residents, streets,” said Dottie Turano. “I know the ridership has increased, that’s why they put in the Select Bus Service line. Spend all that money on actually improving what you got.”

The Feds will rule on the plan by Oct. 1, state officials said.