A man rides his motorcycle at the speed of 50 km/hour. He has to spend Rs 2per km on petrol. If he rides it at a faster speed of 80 km/hour, the petrol cost increases to Rs 3 per km. He has at most Rs 120 to spend on petrol and one hour’s time. He wishes to find the maximum distance that he can travel. Express this problem as a linear programming problem.

Let the man rides his motorcycle for a distance of x km at a speed of 50km/hr then he has to spend Rs. 2/km on petrol.

let the man rides his motorcycle for a distance of y km at a speed of 80 km/hr then he has to spend Rs. 3/km on petrol.

He has at most Rs 120 to spend on petrol for total distance covered so the constraint becomes,

2x+3y≤120…………(i)

Now also given he has at most one hour’s time for total distance to be covered, so the constraint becomes

{as distance=speed×time}

Now taking the LCM as 400, we get

⇒ 8x+5y≤400……………(ii)

And x≥0, y≥0 [non-negative constraint]

He want to find out the maximum distance travelled, here total distance, Z =x+y

Now, we have to maximize the distance, i.e., maximize Z=x+y

So, to maximize distance we have to maximize, Z=x+y, subject to

2x+3y≤120

8x+5y≤400

x≥0, y≥0