Q. 85.0( 2 Votes )

If a young man drives his scooter at a speed of 25 km/hr, he has to spend Rs2 per km on petrol. If he drives the scooter at a speed of 40 km/hour, it produces air pollution and increases his expenditure on petrol to Rs 5 per km. He has a maximum of Rs100 to spend on petrol and travel a maximum distance in one hour time with less pollution. Express this problem as an LPP and solve it graphically. What value do you find here?[CBSE 2013]

Answer :

Let young man drives x km at a speed of 25 km/hr and y km at a speed of 40 km/hr. Clearly,


x, y 0


It is given that, he spends Rs 2 per km if he drives at a speed of 25 km/hr and Rs 5 per km if he drives at a speed of 40 km/hr. Therefore, money spent by him when he travelled x km and y km are Rs 2x and Rs 5y respectively.


It is given that he has a maximum of Rs 100 to spend.


Thus, 2x + 5y 100


Time spent by him when travelling with a speed of 25 km/hr = hr


Time spent by him when travelling with a speed of 40km/hr = hr


Also, the available time is 1 hour.



Or, 40x + 25y1000


The distance covered is Z = x + y which is to be maximized.


Thus, the mathematical formulation of the given linear programming problem is Max Z = x + y subject to


2x + 5y 100


40x + 25y1000


x, y 0


First we will convert inequations as follows:


2x + 5y = 100


40x + 25y = 1000


x = 0 and y = 0.


The region represented by 2x + 5y 100


The line 2x + 5y = 100 meets the coordinate axes at A(50,0) and B(0,20) respectively. By joining these points, we obtain the line 2x + 5y = 100. Clearly (0, 0) satisfies the 2x + 5y = 100. So, the region which contains the origin represents the solution set of the inequation 2x + 5y 100


The region represented by 40x + 25y 1000


The line 40x + 25y = 1000 meets the coordinate axes at C(25,0) and D(0,40) respectively. By joining these points, we obtain the line 2x + y = 12. Clearly (0, 0) satisfies the 40x + 25y = 1000. So, the region which contains the origin represents the solution set of the inequation 40x + 25y 1000


The region represented by x 0, y 0 :


Since every point in the first quadrant satisfies these inequations. So, the first quadrant is the region represented by the inequations x 0 and y 0.


The feasible region determined by the system of constraints


2x + 5y 100, 40x + 25y1000, x 0 and y 0 are as follows


1.jpg


The corner points are O(0,0), B(0,20), E, and C(25,0). The value of Z at these corner points are as follows:



The maximum value of Z is 30 which is attained at E.


Thus, the maximum distance travelled by the young man is 30 kms, if he drives km at a speed of 25 km/hr and km at a speed of 40 km/hr.

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