Is the crown pure gold?

Background

Archimedes was born in 287 BC in a Greek city of Syracuse in what is now Sicily.  He was a mathematician and scientist.  The king of the city, named Hiero II, provided a quantity of pure gold for the fashioning of a laurel wreath made of gold.  It was meant as an offering to the gods of the city.  The goldsmith presented the king with his crown.  It looked magnificent, but to be sure the smith had not kept any of the gold for himself, the king weighed the crown and determined it weighed exactly the same as the pure gold given him for the project.  So the smith was paid.

Later the king was informed that if the smith had replaced a small amount of the pure gold with an equal weight of silver and mixed the gold and silver, the product would look like pure gold.  Not only was the king angered that the smith would cheat him, he worried that if the crown were not pure gold, but it would also offend the gods and bring misfortune to the city. 

So he contacted Archimedes and tasked him with determining if the gold crown was indeed pure gold.  Archimedes knew that pure gold would be denser than gold mixed with silver.  All he had to do was to determine the density of the crown and he would have the king’s answer.  But he had to figure out how to do this without melting down the crown to determine its volume.

The story goes that he was contemplating this problem and decided to think about it while taking a bath. After the tub was full of warm water, he stepped into it and as he lowered himself into the water, the water level rose and some of the water overflowed the tub.  This gave him an idea on how to solve the problem.  Allegedly he was so excited he ran naked through the streets to his study to begin conducting his tests, yelling I have it, which in Greek would be Eureka! Whether this last part is true, the secret to measuring the density of the crown (or any other oddly shaped object) is to submerge it in water.

 

Your Assignment

Using information that would have been available to Archimedes at the time, you are going to perform some simple mass density calculations to determine the validity of his principle.  In other words, you are going to determine for yourself whether or not the crown was made of pure gold. 

Archimedes’ Principle is based upon the approach that if you know the mass of an object, and you can determine the volume of water that it displaces, you can find the density of the object. Today, we calculate density as follows:

LaTeX: Density:=:frac{Mass}{Volume}D e n s i t y = M a s s V o l u m e

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