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Question

If a young man drives his vehicle at 25 km/hr, he has to spend Rs 2 per km on petrol. If he drives it at a faster speed of 40 km/hr, the petrol cost increases to Rs 5/per km. He has Rs 100 to spend on petrol and travel within one hour. Express this as an LPP and solve the same.

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Solution

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,y0

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 is Rs 2x and Rs 5y respectively.

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

Thus, 2x+5y100

Time spent by him when travelling with a speed of 25 km/hr =x25hrTime spent by him when travelling with a speed of 40 km/hr =x40hr

Also, the available time is of 1 hour.

x25+y40140x+25y1000



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

Thus, the mathematical formulat​ion of the given linear programmimg problem is

Max Z =
x+y

subject to

2x+5y10040x+25y1000
x,y0

First we will convert inequations into equations as follows:
2x + 5y = 100, 40x + 25y = 1000, x = 0 and y = 0

Region represented by 2x + 5y ≤ 100:
The line 2x + 5y = 100 meets the coordinate axes at A50, 0 and B0, 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.

Region represented by 40x + 25y ≤ 1000:
The line 40x + 25y = 1000 meets the coordinate axes at C25, 0 and D0, 40 respectively. By joining these points we obtain the line
2x + y = 12.Clearly (0,0) satisfies the inequation 40x + 25y ≤ 1000. So,the region which contains the origin represents the solution set of the inequation 40x + 25y ≤ 1000.


Region represented by x ≥ 0 and 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 + 25y ≤ 1000, x ≥ 0, and y ≥ 0 are as follows



The corner points are O(0, 0), B(0, 20), E503, 403 and C(25, 0).

The values of Z at these corner points are as follows
Corner point Z = x + y
O 0
B 20
E 30
C 25

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 503 km at a speed of 25 km/hr and 403km at a speed of 40 km/hr .





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