How much gold did the goldsmith steal from King Hero's crown?

In summary, Archimedes used the principle of displacement to determine if a goldsmith had stolen any gold from a crown made for King Hero of Syracuse. By measuring the volume of water displaced when the crown was submerged, Archimedes was able to determine that the goldsmith had indeed replaced some gold with silver. The discrepancy in the volume of water displaced indicated that the goldsmith had added silver to the crown, with a resulting total volume of 45.35 mL. Further equations can be written to determine the mass of gold stolen, with one equation for total mass and another for total volume based on the densities of gold and silver.
  • #1
Workout
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Homework Statement


Legend has it that Archimedes first used his discovery to help out his boss King Hero of Syracuse. The king had given 700g of gold to a goldsmith to make him a crown. When the goldsmith brought him the finished crown the King was suspicious that he had defrauded him by replacing some of the gold with silver and keeping the leftover gold. However, the crown still had a mass of 700 g. Archimedes placed the crown in a cylindrical barrel full to the brim with water and collected the water that overflowed. He measured the volume of the displaced water as 42.35cm3 and concluded the goldsmith was guilty. The density of gold is 19.3g/cm3 and the density of silver is 10.5 g/cm3.

(b) What mass of gold had the goldsmith stolen?


Homework Equations


m=dv



The Attempt at a Solution



Ok So I figured out how much water should have been displaced by the gold crown using v=m/d
(0.7kg)/19300kg/m^3

v = 36.27 mL

But the actual amount of water displaced was 42.35 ml. So I know the Vs + Vg = 45.35 mL
Vs = volume of silver, Vg = volume of gold.

I don't know where to go from there...
 
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  • #2
So you know that there's a discrepancy in the volume displaced, and therefor the goldsmith's done something fishy.

Suppose the actual mass of gold used in the crown is mg, and that an amount ms of silver was added. Write an equation for the total mass and another for the total volume (given the densities of gold and silver)...
 

Related to How much gold did the goldsmith steal from King Hero's crown?

What is the story of Archimedes and the Gold Crown?

The story of Archimedes and the Gold Crown is a famous anecdote about the ancient Greek mathematician Archimedes. According to the story, King Hiero II of Syracuse suspected that a gold crown he had commissioned was not made of pure gold. He asked Archimedes to determine the truth without damaging the crown.

What did Archimedes do to solve the problem?

Archimedes was stumped until he took a bath and noticed that the water level rose as he got in. This observation gave him the idea to measure the volume of the crown by submerging it in water. He then compared the volume of the crown to an equal weight of pure gold and found that the crown was less dense, indicating that it was not made of pure gold.

What is the principle behind Archimedes' solution?

The principle behind Archimedes' solution is known as the Archimedes' principle. It states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid that the object displaces. By measuring the volume of the crown and comparing it to an equal weight of pure gold, Archimedes was able to determine the density and composition of the crown.

Did Archimedes' solution prove that the crown was not made of pure gold?

While Archimedes' method did not definitively prove that the crown was not made of pure gold, it provided strong evidence for it. It was later confirmed by melting the crown and measuring its volume, which was found to be larger than that of an equal weight of pure gold. This proved that the crown was likely made of a different material mixed with gold.

Why is the story of Archimedes and the Gold Crown significant?

The story of Archimedes and the Gold Crown is significant because it showcases the brilliance and creativity of Archimedes as a mathematician and scientist. It also demonstrates the practical applications of his work and the lasting impact of his contributions to the field of mathematics and physics.

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