The Water Tank Schedule
A school in Darkhan has a cylindrical water tank of radius 1.5 m and height 4 m. It is filled to 80% of its capacity each morning at 06:00. During the school day, water is consumed at a steady rate of 2.4 m³ per hour from 08:00 to 14:00. A leak at the base loses water at a constant 0.15 m³ per hour at all times. At 14:00, a pump begins refilling the tank at 3 m³ per hour. Use π ≈ 3.14. The school needs at least 6 m³ of water remaining at 18:00 for evening cleaning.
1. Calculate the water volume in the tank at 14:00 and determine whether the tank will meet the 18:00 requirement.
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Total capacity = πr²h = 3.14 × 2.25 × 4 = 28.26 m³. Initial volume at 06:00 = 0.8 × 28.26 = 22.608 m³. From 06:00 to 08:00 (2 h), only the leak acts: loss = 0.15 × 2 = 0.3 m³. Volume at 08:00 = 22.308 m³. From 08:00 to 14:00 (6 h): consumption = 2.4 × 6 = 14.4 m³; leak = 0.15 × 6 = 0.9 m³. Total loss = 15.3 m³. Volume at 14:00 = 22.308 − 15.3 = 7.008 m³. From 14:00 to 18:00 (4 h): pump adds 3 × 4 = 12 m³; leak removes 0.15 × 4 = 0.6 m³. Net gain = 11.4 m³. Volume at 18:00 = 7.008 + 11.4 = 18.408 m³, which exceeds 6 m³. The requirement is met with a large margin.
2. The leak worsens, doubling to 0.30 m³/h at all times. Recalculate the 18:00 volume and state whether the requirement is still met.
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Recompute with leak = 0.30 m³/h. 06:00–08:00: loss = 0.6 m³ → 22.008 m³ at 08:00. 08:00–14:00: consumption 14.4 + leak 1.8 = 16.2 m³ → volume at 14:00 = 22.008 − 16.2 = 5.808 m³. 14:00–18:00: pump 12 − leak 1.2 = net +10.8 → volume at 18:00 = 5.808 + 10.8 = 16.608 m³. Still above 6 m³, so the requirement is met. The doubled leak reduces the final volume by 1.8 m³ compared with the original case but does not threaten the threshold.
3. The pump can only run for 2 hours (14:00–16:00) instead of 4, and the leak is back to 0.15 m³/h. Does the school still have 6 m³ at 18:00? Justify with numbers.
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Volume at 14:00 (original leak) = 7.008 m³. 14:00–16:00: pump adds 6 m³, leak removes 0.3 m³ → +5.7 → 12.708 m³ at 16:00. 16:00–18:00: no pump; leak removes 0.3 m³ → 12.408 m³ at 18:00. This is above 6 m³, so the requirement is met. Even with only half the pumping time, the margin is comfortable because the 14:00 volume was still positive.