Pipes and Cisterns — Mains Questions

A water treatment plant has three inlet pipes with different capacities and two outlet pipes for maintenance. The inlet pipes can individually fill the main reservoir in 8, 12, and 24 hours respectively. The outlet pipes can individually empty the reservoir in 16 and 48 hours respectively. Analyze the optimal operational strategy for filling the reservoir in minimum time while ensuring at least one outlet pipe remains operational for safety. Calculate the time required under this optimal strategy and justify your approach.

A smart city project involves designing an automated water distribution system where multiple tanks are filled simultaneously through a network of pipes with varying efficiencies. Tank A can be filled by pipes with rates in ratio 3:4:5, Tank B by pipes with rates in ratio 2:3:7, and both tanks have emergency drain pipes. If the total system must maintain 80% efficiency during peak hours while handling potential pipe failures, design a mathematical model to ensure continuous water supply. Evaluate the system's resilience and calculate backup requirements.

An agricultural irrigation project requires filling multiple field reservoirs using a combination of tube wells and canal water supply. The tube wells have varying capacities due to different water table levels, while canal supply is subject to scheduled releases. Design an integrated water management strategy that optimizes filling times across all reservoirs while considering water conservation, energy costs, and crop irrigation schedules. Analyze the trade-offs between efficiency and sustainability in this multi-constraint optimization problem.