Have you heard of the ancient Greek scholar Archimedes and his theory of buoyancy? Do you know the story of how he came up with his principle? Apparently, it all started when Hiero II, the king of Syracuse at the time, gave Archimedes a golden crown and asked him to assess its purity.

#### Eureka!

The king had given his goldsmith a certain amount of pure gold with which to craft a crown. The goldsmith presented him with a crown of the same weight, but the king suspected that the goldsmith had removed some gold to keep for himself and substituted an equivalent weight of silver. Hiero wanted the crown to be tested for its purity but kept intact.

While pondering the king’s problem, Archimedes thought up a way to precisely measure the volume of irregular objects, realizing that the volume of water displaced by an object had to be equal to the volume of the object. He stumbled across this insight when he stepped into a bath and noticed the resulting rise in water level. He reportedly proclaimed, "Eureka!" and ran out to the streets of Syracuse, naked in his utter excitement.

Archimedes measured the volume of the crown by inserting it in water. He concluded that the goldsmith had indeed replaced some of the gold with silver because the total volume of the crown was greater than the volume of the original amount of gold, which was due to the fact that gold has a higher density than silver.

You can read a more detailed version of the legend here:

http://www.longlongtimeago.com/llta_greatdiscoveries_archimedes_eureka.html

#### Explore More

Now let’s examine the role of linear equations in this purity assessment. Let’s assume that the king gave 500 grams of gold to the goldsmith, and the goldsmith crafted a 500-gram crown. Archimedes determined the volume of the crown to be 9050 cubic centimeters, and he knew that densities of gold and silver are 19 g/cm^{3} and 10 g/cm^{3}, respectively. How many grams of silver, if any, were used to replace gold in the crown?