Question

If John drives a car at a speed of 60 km/hour, he has to spend ₹ 5 per km on petrol. If he drives at a faster speed of 90 km/hour, the cost of petrol increases ₹ 8 per km. He has ₹ 600 to spend on petrol and wishes to travel the maximum distance within an hour. Formulate the above problem as L.P.P.

Answer 1

Let John travel x₁ km at speed of 60 km/hr and x₂ km at a speed of 90 km/hr.
∴  Total distance = (x₁ + x₂) km

Time = `"Distance"/"Speed"`

Time to travel x1 km = `(x_1/60)` hours and time to travel x₂ km = `(x_2/90)` hours.

∴ Total time = `(x_1/60 + x_2/90)"hours"`
But John wishes to travel maximum distance within an hour.

∴ `x_1/(60) + x_2/(90) ≤ 1`
John has to spend ₹ 5 per km at 60 km/hr and ₹ 8 per km at 90 km/hr.
∴ Total cost = ₹ (5x₁ + 8x₂)
But John has ₹ 600 to spend on petrol
∴ 5x₁ + 8x₂ ≤ 600
Since x₁ and x₂ cannot be negative, we have x₁ ≥ 0, x₂ ≥ 0
∴ Given problem can be formulated as follows:
Maximize Z = x1 + x2,

Subject to `x_1/(60) + x_2/(90) ≤1, 5x_1 + 8x_2 ≤ 600`, x₁ ≥ 0, x₂ ≥ 0.