Algebra 2 (Linear Programming)
The Carbon coal company has 2 mines, a surface mine ad a deep mine. it costs $200 per day to operate the surface mine and $250 to operate the deep mine. Each mine produces a medium grade and a medium-hard grade coal, but in different proportions. This surface mine produces 12 tons of medium grade and 6 tons of medium-hard grade coal per day, and the deep mine produces 4 tons of medium grade and 8 tons of medium-hard grade coal every day. The company has a contract to deliver at least 600 tons of medium grade and 480 tons of medium-hard grade coal within 60 days. How many days should each mine be operated so that the contract can be filled at minimum cost?
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The Carbon Coal Company can operate the surface mine for 30 days and the deep mine for 28 days to fulfill the contract at minimum cost. Operating the surface mine for 30 days produces 360 tons of medium grade coal and 180 tons of medium-hard grade coal and operating the deep mine for 28 days produces 112 tons of the medium grade coal and 224 tons of the medium-hard grade coal, which together produce 472 tons of the medium grade coal and 404 tons of the medium-hard grade coal, totaling 876 tons, which is more than what is needed to fulfill the contract. The total cost for this operation is 30 X $200 + 28 X $250 = $10,200, which is the minimum possible cost for fulfilling the contract.
