Tap A fills a tank in 4 hours whereas tap B empties the full tank in 24 hours. A and B are opened alternately for 1 hour each. Every 2 hours the level of water is found to increase by 0.5 m. The depth of the tank is

### Concept & Proportionality of Volume and Depth In uniform tanks, the fraction of the tank filled is directly proportional to the fraction of the total depth filled. We analyze the net work done in one complete alternate cycle. $$ \text{Fraction Filled} \times \text{Total Depth} = \text{Observed Depth Increase} $$ ### Step-by-Step Solution 1. **Given**: Tap A fills in 4 hours, Tap B empties in 24 hours. They run alternately. 2. **Calculation**: - In 1 hour, Tap A fills 1/4 of the tank. - In 1 hour, Tap B empties 1/24 of the tank. - Since they operate alternately, a full cycle takes 2 hours. - Net part of the tank filled in 2 hours = (1/4) - (1/24) - Make denominators equal: 1/4 = 6/24. - Net filled = 6/24 - 1/24 = 5/24 of the tank. 3. Relate volume to depth: - We are given that in 2 hours, the water level increases by 0.5 m. - Therefore, 5/24 of the total depth ($D$) corresponds to 0.5 m. - $(5/24) \times D = 0.5$ 4. Solve for Depth ($D$): - $D = 0.5 \times (24 / 5)$ - $D = 1/2 \times 24 / 5 = 12 / 5 = 2.4$ meters. ### Exam Strategy & Shortcut Recognize that 2 hours is exactly one cycle. Net fill per cycle is $1/4 - 1/24 = 5/24$. If $5/24$ of the tank is $0.5$ meters, then $1/24$ of the tank is $0.1$ meters. The full tank ($24/24$) is $24 \times 0.1 = 2.4$ meters. ### Common Pitfall A common error is confusing the alternate hour setup and accidentally calculating the net rate assuming they run simultaneously for a full 2 hours, which leads to incorrect capacity ratios. ### Final Answer Therefore, the correct answer is **2.4 m**.