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

[Workout]

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...

 

[gneill]

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 m_g, and that an amount m_s of silver was added. Write an equation for the total mass and another for the total volume (given the densities of gold and silver)...