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?

### Concept & Logic This problem models a tank with a constant initial volume and a continuous, constant inflow. To ensure the supply never fails, the daily usage must exactly equal the daily inflow rate. If usage exceeds inflow, the initial reserve depletes over time. Let initial volume = $V$ and daily inflow = $r$. Total volume used over $d$ days = $V + d \times r$. ### Step-by-Step Solution 1. Let $V$ be the initial capacity of the tank and $r$ be the rate of constant inflow per day. 2. Case 1: 32000 gallons used daily for 50 days. $$V + 50r = 32000 \times 50$$ $$V + 50r = 1600000 \text{ --- (Equation 1)}$$ 3. Case 2: 37000 gallons used daily for 40 days. $$V + 40r = 37000 \times 40$$ $$V + 40r = 1480000 \text{ --- (Equation 2)}$$ 4. Subtract Equation 2 from Equation 1 to eliminate $V$: $$(V + 50r) - (V + 40r) = 1600000 - 1480000$$ $$10r = 120000$$ $$r = 12000 \text{ gallons/day}$$ 5. For the supply to never fail, the daily usage must not exceed the daily inflow. Thus, the maximum sustainable usage is equal to the inflow rate. $$\text{Maximum usage} = r = 12000 \text{ gallons}$$ ### Exam Strategy & Shortcut Focus on the differences. The difference in total water consumed is due to the difference in inflow over those extra days. Total consumed in 50 days = 1600k Total consumed in 40 days = 1480k Difference = 120k gallons. This extra 120k came from 10 extra days of inflow. Inflow rate = $120k / 10 = 12k$ gallons/day. This is the sustainable rate. ### Common Pitfall A common mistake is trying to solve for $V$ and then getting confused about how to apply it. The key is realizing that "never failing" means operating strictly on the renewable inflow. ### Final Answer Therefore, the correct answer is **12000 gallons**.