Mountain Laurel Vineyards produces three kinds of wine—Mountain Blanc, Mountain Red, and Mountain Blush. The company has 17 tons of grapes available to produce wine this season. A cask of Blanc requires 0.21 ton of grapes, a cask of Red requires 0.24 ton, and a cask of Blush requires 0.18 ton. The vineyard has enough storage space in its aging room to store 80 casks of wine.
The vineyard has 2,500 hours of production capacity, and it requires 12 hours to produce a cask of Blanc, 14.5 hours to produce a cask of Red, and 16 hours to produce a cask of Blush. From past sales the vineyard knows that demand for the Blush will be no more than half of the sales of the other two wines combined. The profit for a cask of Blanc is $7,500, the profit for a cask of Red is $8,200, and the profit for a cask of Blush is $10,500.
Formulate a linear programming model for this problem.