BANGOR, Maine — After two last-minute amendments failed, city councilors on Monday narrowly approved a 2010-11 municipal budget that will increase the tax rate by 15 cents for every $1,000 of property value.

The amendments — one to reduce overtime for the Police Department and to defund an economic development position, the other to use undesignated reserve funds to offset the tax increase — both failed by 5-4 votes, representing a sharp philosophical divide among councilors.

In the end, the $45.5 million budget that also passed by a 5-4 vote seemed to please none of the councilors.

The new tax rate of $19.20 per $1,000 of property valuation translates to an increase of $30 a year on a $200,000 home. Six cents of the increase comes from the municipal side, 5 cents from the school department’s share and 4 cents is represented by an increase in Bangor’s share to Penobscot County.

Monday’s council meeting and final vote culminated months of budget workshops, discussions and public sessions, but the final-hour amendments brought back to light some of the frustration councilors felt during the lengthy process.

The first amendment, pitched by Councilor Hal Wheeler, asked the council to trim $40,000 in overtime from the Police Department and cut $40,000 that was to fund an economic development office position. Wheeler and others said the cuts were not out of the blue and represented a last good-faith effort to trim a few more dollars.

Others disagreed.

“To slap this on at the last minute is unfair,” Councilor Gerry Palmer said.

“There were some cuts we asked for that didn’t come as close as I wanted, but this amendment goes against an agreement we already made,” added Councilor Cary Weston.

The final vote on that amendment featured Weston, Palmer, Pat Blanchette, Geoff Gratwick and Susan Hawes opposing the suggestion. Wheeler, David Nealley, Richard Bronson and Chairman Richard Stone favored the idea.

The second amendment, proposed by Nealley, proved even more heated. Nealley suggested using $360,000 from the city’s undesignated fund balance to offset the tax increase. Every nickel on the tax rate represents $120,000 of budget, so the $360,000 would have brought the proposed 2010-11 tax rate back down to $19.05, where it sits today.

Nealley said if the city approved a tax rate of $19.20, that would become the baseline, making it harder to ever bring that rate back down below $19.20.

Palmer and others cautioned against robbing the city’s piggy bank.

“The undesignated fund is like a narcotic,” he said. “You become addicted and then, before you know it, it’s gone.”

Wheeler chastised Palmer and others for a doomsday attitude.

“The money belongs to the taxpayers; we should give them a break,” he said.

After significant debate, another narrow vote defeated the amendment. Nealley, Weston, Wheeler and Bronson favored using undesignated funds to hold the tax rate flat. Palmer, Blanchette, Hawes, Gratwick and Stone struck the measure down.

Both sides seemed unhappy to even discuss the amendment.

“We should have held the tax rate through the operating budget,” Bronson said, a sentiment that Hawes echoed moments later.

Even before Monday, the 2010-11 municipal budget discussion featured a number of controversial proposals that ultimately failed to win support of a majority of councilors. Among those failed ideas were 10 percent cuts to the Fire and Public Works departments and a reduction of $170,000 in funding to the Bangor Public Library.

In other noteworthy budget items, councilors agreed not to cut holiday and Saturday service on one of the city’s bus routes but did agree to increase fares from $1 to $1.25. Councilors also agreed not to increase fees for the Beth Pancoe Municipal Aquatic Center or establish them at Dakin Pool, which originally were proposed.

The new budget goes into effect on July 1.