A mining company owns two mines These mines produce an ore t

A mining company owns two mines. These mines produce an ore that can be graded into two classes: regular grade and low grade. The company must produce at least 410 tons of regular-grade and 530 tons of low-grade ore per week. The first mine produces 6 tons of regular-grade and 16 tons of low-grade ore per hour. The second mine produces 20 tons of regular-grade and 10 tons of low-grade ore per hour. The operating cost of the first mine is $9000 per hour, and the operating cost of the second mine is $30,500 per hour. The first mine can be operated no more than 55 hours a week, and the second mine can be operated no more than 29 hours a week. How many hours per week should each mine be operated to minimize the cost?

Solution

Let the 1st and the 2nd mine be operated for x and y hours per week respectively. Then x 55 and y 29. Also, 6x + 20 y = 410 or, 3x + 10 y = 205...(1) and 16x + 10y = 530...(2) We have find a solution for these 2 equations, within the constraints x 55 and y 29. On subtracting the 1st equation from the 2nd equation, we get 16x + 10 y -3x - 10y = 530 -205 or, 13x = 325 so that x = 325/13 = 25. Now, on substituting x = 25 in the 1st equation, we get 3*25 + 10 y = 205 or, 10 y = 205 - 75 = 130, so that y = 130/10 = 13. These values of x and y satisfy both the given constraints. Thus,  to minimize the cost, the 1st and the 2nd mines should be operated for 25 and 13 hours respectively in a week.