Question

two mining fields, field a and field b, of a coal mining company produce lignite and bituminous coal. the operating cost per day for field a and field b are $55,000 and $45,000, respectively. the recent records at the company indicate that field a can produce 250 tons of lignite along with 300 tons of bituminous coal per day, whereas field b can produce 200 tons of lignite along with 450 tons of bituminous coal per day. the expected demands to be met are 120,000 tons of lignite and 170,000 tons of bituminous coal. to minimize the operating costs of the mining fields, how many days does the company need to operate each of these fields?

Answer 1

Let x be the number of days that Field A needs to operate and y be the number of days that Field B needs to operate.

We can set up the following system of equations based on the information given:

250x + 200y = 120,000 (tons of lignite)
300x + 450y = 170,000 (tons of bituminous coal)

To minimize the operating costs, we need to minimize the total cost of operating both fields, which is given by:

Total Cost = 55,000x + 45,000y

We can use the system of equations to solve for x and y, and then substitute into the total cost equation to get the minimum cost.

Multiplying the first equation by 3 and the second equation by -2, we can eliminate the x term:

750x + 600y = 360,000
-600x - 900y = -340,000

Adding the two equations, we get:

150x = 20,000
x = 133.33

Substituting into the first equation, we get:

250(133.33) + 200y = 120,000
y = 235

Therefore, the company needs to operate Field A for 133.33 days (rounded up to 134 days) and Field B for 235 days to minimize the operating costs.