A single reservoir supplies petrol to an entire city and is fed continuously by one pipeline that fills it with a uniform inflow rate. When the reservoir is full and 40,000 litres of petrol are used every day, the supply runs out in 90 days. If only 32,000 litres of petrol are used daily, the supply fails in 60 days. How many litres of petrol can be used per day so that the supply never fails in the long run?

Introduction / Context:
This is a reservoir and uniform inflow problem based on time and work concepts. The reservoir is being filled continuously by a pipeline while petrol is simultaneously used up. We are given two different daily usage levels and the durations until the reservoir becomes empty, and we must determine the maximum sustainable daily usage that will never exhaust the supply in the long run.

Given Data / Assumptions:

• Let the capacity of the reservoir be C litres.
• Let the daily inflow from the pipeline be R litres per day (constant).
• Case 1: Daily usage = 40000 litres; the reservoir lasts for 90 days.
• Case 2: Daily usage = 32000 litres; the reservoir lasts for 60 days.
• We assume the reservoir is full at the start of each scenario and that inflow and outflow are continuous and uniform.

Concept / Approach:
The change in the amount of petrol stored equals inflow minus outflow. Over the entire duration until the reservoir becomes empty, the net amount used equals the initial capacity plus any extra inflow. We can express this by linear equations in terms of C and R. Solving these simultaneous equations gives the daily inflow R. If the city uses exactly R litres per day, then inflow equals outflow, so the water level remains constant and the supply never fails.

Step-by-Step Solution:
For Case 1: C + 90 * R - 90 * 40000 = 0, so C + 90 R = 90 * 40000. For Case 2: C + 60 * R - 60 * 32000 = 0, so C + 60 R = 60 * 32000. Subtract the second equation from the first: (C + 90 R) - (C + 60 R) = 90 * 40000 - 60 * 32000. This simplifies to 30 R = 90 * 40000 - 60 * 32000. Compute the right side: 90 * 40000 = 3600000 and 60 * 32000 = 1920000, so 30 R = 1680000. Therefore, R = 1680000 / 30 = 56000 litres per day. If the city uses petrol at exactly 56000 litres per day, the daily outflow equals the inflow from the pipeline, so the reservoir level never decreases and supply does not fail.

Verification / Alternative check:
You can use the found inflow R = 56000 litres to recover C from either of the two equations and check that the reservoir indeed becomes empty at the stated times when daily usage is 40000 or 32000 litres. Both equations will yield the same value of C, confirming that the inflow value is consistent with both scenarios. This validates the correctness of R = 56000 litres per day.

Why Other Options Are Wrong:
64000 litres is larger than the actual inflow, so long term usage at that rate would drain the reservoir. 78000 litres is even larger and clearly unsustainable compared with the inflow. 60000 litres also exceeds 56000 litres, so the reservoir would slowly empty. 52000 litres is less than the inflow, so it is safe, but the question asks for the maximum daily usage without failure, which is the inflow rate itself.

Common Pitfalls:
Students sometimes forget that inflow continues while the reservoir is being used, so they treat the problem as a simple capacity divided by consumption case. Another mistake is mixing up units or miscalculating the subtraction step when forming the simultaneous equations. Always clearly define C and R, write equations for each scenario, and subtract carefully to remove C and solve for R.

Final Answer:
The maximum petrol that can be used daily without the supply ever failing is 56000 litres per day.