How did the sinking of the lusitania lead to us entry into the war in europe?

the prevailing outlook called for a quick resolution to hostilities.

the military had developed new weapons to employ.

the loss of american life was too serious to ignore.

the american public wanted revenge for the assassination of franz ferdinand.

Correct

Public outrage in the US against the Germans swept the nation following the sinking of the British ocean liner, Lusitania.  When a German U-boat (submarine) sank the Lusitania in May, 1915, over 1,000 persons were killed, including more than 100 Americans. The passenger liner was targeted by the Germans because they suspected weapons were being shipped to Britain in the cargo hold of the ship.  Germany managed to stave off American entry into the war at the time by pledging to stop submarine attacks.  

A couple years later, however, Germany resumed unrestricted submarine warfare.  There was also an intercepted telegram (the "Zimmerman Telegram") that showed Germany was trying to secure Mexico as an ally against the United States.  Those events in 1917 led the US to declare war on Germany in response.

Here's an an example of American feeling at the time of the Lusitania incident.  Gifford Pinchot, who had been the Chief of the US Forestry Service (from 1905 to 1910) , was quoted in the New York Times in May, 1915, after he had just recently returned from Europe. He asserted that Americans on the Lusitania (along with other passengers) were killed because an autocratic military empire was trying to dominate nations that were self-governing. His characterization of German intentions mirrored how President Woodrow Wilson later called on the USA to enter the war to "make the world safe for democracy."

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