A town is supplied with water from a big overhead tank which is fed with a constant volume of water regularly. When the tank is full, if 32000 gallons are used daily, the supply fails in 50 days. However, if 37000 gallons are used daily, the supply lasts for 40 days only. How much water can be used daily without the supply ever failing?
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This is a classic "inflow and outflow" problem. The difference in total water used over different time periods gives the total inflow over the difference in days, allowing you to solve for the daily inflow rate.
Step 1: Understanding the Concept:
This is a standard "Inflow-Outflow" problem where the tank empties because outflow exceeds inflow. The supply will never fail if outflow \( \le \) inflow. Step 2: Key Formula or Approach:
Initial Volume + (Inflow Rate \( \times \) Time) \( = \) Outflow Rate \( \times \) Time. Step 3: Detailed Explanation:
Let \( V \) be the initial full volume and \( I \) be the daily inflow.
Case 1: \( V + 50I = 50 \cdot 32000 = 1600000 \). (i)
Case 2: \( V + 40I = 40 \cdot 37000 = 1480000 \). (ii)
Subtracting (ii) from (i):
\( 10I = 1600000 - 1480000 = 120000 \).
\( I = 12000 \) gallons per day.
If the daily usage equals the constant inflow rate, the tank volume will never drop. Step 4: Final Answer:
12000 gallons can be used daily.