How was the annexation of Hawaii vital to the establishment of the United State's
sphere of influence in the Pacific ?

The United States wanted Hawaii to acquire its islands and because it was a port way to China, East India and Asia. Not only did they want the islands, but they wanted their naval base. They wanted their naval base for war so they would have another advantage to help defeat the other country.

answer; william penn's approach made the colony diverse by attracting many types of people and religions;