What did archimedes discover in the bath? why was he so excited? describe the discovery using geometric terms. what do you predict archimedes did next with his new discoveries? explain.

Eureka!

He discovered that the amount of liquid displaced when an object is submerged in the liquid is the *same* as the one of the object. It does not need to sink. For instance, Archimedes got in the bath tub and the water spilled off, he thought Eureka! Why?

The King of Syracuse wanted to know whether a crown forged for himself was all made of gold or had been mixed with something else. He did not want to break the beautiful crown. No one knew.

Archimedes went back to palace (the legend says naked because of his excitement), put the crown in a barrel with water, measure the volume of water displaced, and derive the density of the crown (having measured its weight). the density was different than the one of gold.

The king broke the crown and indeed found it was an alloy! The goldsmith was ... killed :(

Ican't do any of the other stuff, but i can get you started. the answer, i believe, is $2.50.