About 2,200 years ago, one of the most famous mathematicians in history made a major discovery. The mathematician was Archimedes, who was presented with a problem by King Hieron II of Syracuse in Sicily. The king had given a bar of gold to a jeweler and ordered him to make it into a crown. After receiving the crown, the king suspected that the jeweler had replaced some of the gold with an equal weight of a cheaper metal like silver and kept the remainder of the gold for himself. The king had no way to determine whether this was true, so he gave the crown to Archimedes and asked him to devise a way to find out.

At the time, Archimedes knew that gold was more dense than silver. So, if the volume of the crown was greater than the volume of a bar of gold, they would have proof that the jeweler had stolen some of the gold. However, Archimedes had no way to find the volume of an irregular shape like a crown.

Archimedes struggled with this problem for a long time. One day, he went to take a bath. As he lowered himself into the bath, he noticed that the water level rose. As he continued to lower himself into the water, the water level continued to rise. He was so excited at his discovery that he jumped out of the bath and, before he could remember to put his pants on, went running through the streets yelling “Eureka! Eureka!” which in Greek means “I’ve found it! I’ve found it!” He now had a way to measure the volume of the irregularly shaped crown.

In Archimedes’s case, the jeweler had indeed stolen gold from the king. This did not end well for the jeweler.

Part A
A key component to this story is that the jeweler deceptively replaced “an equal weight” (or equal mass) of gold with silver. How does this action result in increasing the volume of the crown? Explain using one or more equations.
Part B
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.