Question
If a young man drives his vehicle at 25 km/hr, he has to spend ₹2 per km on petrol. If he drives it at a faster speed of 40 km/hr, the petrol cost increases to ₹5 per km. He has ₹100 to spend on petrol and travel within one hour. Express this as an LPP and solve the same.
Solution
Let us assume that the man travels x km when the speed is 25 km/hour and y km when the speed is 40 km/hour.
Thus, the total distance travelled is (x + y) km.
Now, it is given that the man has Rs 100 to spend on petrol.
Total cost of petrol = 2x + 5y ≤ 100
Now, time taken to travel x km = \[\frac{x}{25}\] h Time taken to travel y km = \[\frac{y}{40}\] h Now, it is given that the maximum time is 1 hour. So,
\[ \Rightarrow 8x + 5y \leq 200\]
Maximise Z = x + y
subject to the constraints
2x + 5y ≤ 100
8x + 5y ≤ 200
x ≥ 0, y ≥ 0
The feasible region determined by the given constraints can be diagrammatically represented as,
The coordinates of the corner points of the feasible region are O(0, 0), A(25, 0), B
The value of the objective function at these points are given in the following table.
Corner Points  Z = x + y 
(0, 0)  0 + 0 = 0 
(25, 0)  25 + 0 = 25 
\[\left( \frac{50}{3}, \frac{40}{3} \right)\]

\[\frac{50}{3} + \frac{40}{3} = 30\]

(0, 20)  0 + 20 = 20 
So, the maximum value of Z is 30 at \[x = \frac{50}{3}, y = \frac{40}{3}\] Thus, the maximum distance that the man can travel in one hour is 30 km.
Hence, the distance travelled by the man at the speed of 25 km/hour is \[\frac{50}{3}\] km, and the distance travelled by him at the speed of 40 km/hour is \[\frac{40}{3}\] Km.