1
Question
An oil company required 13000, 20000 and 15000 barrels of high grade, medium grade and low grade oil respectively. Refinery A produces 100, 300 and 200 barrels per day of high grade, medium grade and low grade oil respectively. While, refinery B produces 200, 400 and 100 barrels per day of high grade, medium grade and low grade oil respectively. If refinery A costs Rs 400 per day and refinery B costs Rs 300 per day to operate, then the days should each be run to minimize costs, while satisfying requirements are
An oil company required 13000, 20000 and 15000 barrels of high grade, medium grade and low grade oil respectively. Refinery A produces 100, 300 and 200 barrels per day of high grade, medium grade and low grade oil respectively. While, refinery B produces 200, 400 and 100 barrels per day of high grade, medium grade and low grade oil respectively. If refinery A costs Rs 400 per day and refinery B costs Rs 300 per day to operate, then the days should each be run to minimize costs, while satisfying requirements are
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Solution
The correct option is B 60,30
Let days of refinery A be x and days of refinery B be y.
Total costs incurred will be equal to 400x+300y
1. High Grade barrel
Production by Refinery A in x days = 100xProduction by Refinery B in y days= 200yTotal minimum required production = 12000
2. Medium grade barrel
Production by Refinery A in x days = 300x
Production by Refinery B in y days= 400yTotal minimum required production = 20000
3. Low grade barrel
Production by Refinery A in x days = 200xProduction by Refinery B in y days= 100yTotal minimum required production = 15000
Therefore, the problem statement when written in mathematical form is that, we need to minimise the total cost i.e 400x+300y keeping in mind that minimum requirement of the factory is met, i.e
100x+200y⩾12000300x+400y⩾20000
200x+100y⩾15000
If we refer the graph,shaded region is the required area of consideration and there are three points on which function needs to be checked
1. (0,150)2. (60,30)3. (120,0)
The value of the function 400x+300y is minimum when x=60,y=30.
Let days of refinery A be x and days of refinery B be y.
Total costs incurred will be equal to 400x+300y
1. High Grade barrel
Production by Refinery A in x days = 100x
Production by Refinery B in y days= 200y
Total minimum required production = 12000
2. Medium grade barrel
Production by Refinery A in x days = 300x
Production by Refinery B in y days= 400y
Total minimum required production = 20000
3. Low grade barrel
Production by Refinery A in x days = 200x
Production by Refinery B in y days= 100y
Total minimum required production = 15000
Therefore, the problem statement when written in mathematical form is that, we need to minimise the total cost i.e 400x+300y keeping in mind that minimum requirement of the factory is met, i.e
100x+200y⩾12000
300x+400y⩾20000
200x+100y⩾15000
If we refer the graph,shaded region is the required area of consideration and there are three points on which function needs to be checked
1. (0,150)
2. (60,30)
3. (120,0)
The value of the function 400x+300y is minimum when x=60,y=30.
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