1. The problem statement, all variables and given/known data An alloy of gold, aluminum, and copper has a density of 10,000 kg/m3. The alloy contains at least 10% aluminum and 5% copper by mass. The densities for the three metals are respectively ρAu = 19320 kg/m3, ρAl = 2712 kg/m3, ρCu = 8940 kg/m3. Find the maximum and minimum percentage of gold by mass. Show all work and carefully explain your logic. 2. Relevant equations ρ=Mass/Volume Lagrange Multipliers 3. The attempt at a solution I am struggling to form the initial functions. I suspect the professor intends for us to use Lagrange multipliers, and I am confident I can find the gradients and drudge through the resulting algebra, but I am struggling to get started. I understand density is mass per volume, and that percent by mass is the mass of a component divided by the total mass. I imagine I will take the three metals to be the variables x, y, and z. And I suspect optimizing over a single cubic meter would be the best approach. I found some guidance online suggesting I use the sum of the masses and the sum of the volumes. This gives me: MAu = ρAu * VAu MAl = ρAl * VAl MCu = ρCu * VCu MAu + MAl + MCu = 10,000 kg ρAu * VAu + ρAl * VAl + ρCu * VCu = 10,000 kg 19320VAu + 2712VAl + 8940VCu = 10,000 kg and VAu + VAl + VCu = 1 m3 Both of these seem to be constraint equations and I am confused as to what the function to be optimized will look like. Other constraints would be: (assuming optimization over a single cubic meter) MCu ≥ 500 kg MAl ≥ 1000 kg MAu ≤ 8500 kg The Lagrange Multiplier method requires two functions. One function constrained by another function. I have found help here before, but this is my first post. I would appreciate any insight regarding the initial functions. Thank you.