Archimedes used density along with mass and volume of alloys (mixtures) of gold and silver to determine the percent of gold (and also silver) in these alloys by simple, non-destructive measurements.
Prove that the mass of gold in a given mass of silver alloy is given by the equation:
MAu= (Malloy-Valloy PAg) * PAu/(PAu-PAg)
You will need to set up two simultaneous equations in two unknowns that include the densities of sliver and gold, and the weight and volume of the alloy!
To prove the given equation, we can start by setting up two simultaneous equations based on the given information.
Let's assume we have a silver alloy with mass Malloy and volume Valloy. We want to find the mass of gold in this alloy, denoted as MAu.
Equation 1: Density of the alloy (D) = (Mass of the alloy) / (Volume of the alloy)
D = (Malloy + MAu) / Valloy
Next, we can express the density of the alloy in terms of the densities of gold (DAu) and silver (DAg) and their respective masses and volumes. We will use the equation:
D = (Volume of gold) * (Density of gold) + (Volume of silver) * (Density of silver)
D = MAu / (PAu) + (Malloy - MAu) / (PAg)
Here, PAu represents the density of gold, and PAg represents the density of silver.
Now, we can rearrange equation 1 to solve for MAu:
D * Valloy = Malloy + MAu
Substitute D*Valloy into the expression for D in equation 2:
(Malloy + MAu) = MAu / (PAu) + (Malloy - MAu) / (PAg)
Now, we can simplify this equation by cross-multiplying and combining like terms:
Malloy * (PAg) * (PAu) + MAu * (PAg) = MAu * (PAu) + Malloy * (PAu)
Multiply through by (PAu - PAg) to eliminate the denominators:
Malloy * (PAg) * (PAu) + MAu * (PAg) * (PAu - PAg) = MAu * (PAu) * (PAu - PAg) + Malloy * (PAu) * (PAu - PAg)
Expand and simplify:
Malloy * (PAg) * (PAu) + MAu * (PAg) * PAu - MAu * (PAg)^2 = MAu * (PAu)^2 - MAu * (PAg) * (PAu) + Malloy * (PAu) * (PAu) - Malloy * (PAu) * (PAg)
Move all terms involving MAu to one side of the equation:
MAu * (PAg) * (PAu - (PAg)^2 + (PAg) * (PAu) - (PAu)^2) = Malloy * (PAu) * ((PAu) - (PAg))
Combine like terms:
MAu * (PAg) * ((PAu) - (PAg)) = Malloy * (PAu) * ((PAu) - (PAg))
Divide both sides by (PAg) * ((PAu) - (PAg)):
MAu = Malloy * (PAu) / ((PAu) - (PAg))
Therefore, the mass of gold in a given mass of silver alloy is determined by the equation:
MAu = (Malloy * PAu) / (PAu - PAg)
This equation allows us to find the mass of gold in a silver alloy using density measurements along with mass and volume measurements.