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Question 04  

A refining company must produce 70 oz. of gold and 500 oz. of silver from ore. The ores available for purchase have the following characteristics.
How many tons of each type of ore should be purchased in order to produce the metals at least cost? 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

Suppose you order a amount of type A, b of type B and c of type C. What is your objective function that you want to optimize? 
Hint 2 

Suppose you order a amount of type A, b of type B and c of type C. How large is your gold yield? How large is your silver yield? Can you write this in matrix notation? 
Hint 3 

Remember that you can not order negative quantities. 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Let's say we order a amount of type A, similarly b of type B and c of type C. Thus, the total cost can be read as which is the objective function we must minimize. The variables are not arbitrary, but must be chosen so that we get 70oz of gold and 500oz of silver. A choice of a,b,c leads to two equations relating to the amount of ore purchased: Amount of gold: Amount of silver: Put this into an extended matrix we get:
Which we can simplify to the matrix:
Thus, we can consider c to be a free variable. Solving this system we get Having written each quantity with respect to c we can put it in the cost equation which we now have to minimize. C is an increasing function (it is linear with positive slope) and so choosing the smallest value of c possible will yield the minimum. It looks like this value may be c=0 however looking at the other formulas tells us that if c were zero then a would be 20 and b would be 90. We can't order negative quantities of anything so we actually have hidden constraints, namely that Using our relations for a and b, this tells us that Since we know the cost C is an increasing function, we want to take the smallest c allowable which is c = 20. Using the equations above we find that this yields a = 0 and b = 30. 