A cylindrical tank with a radius of 3 meters and a height of 10 meters is filled with water. If the water level drops by 0.5 meters per hour due to a leak, how long will it take for the tank to be completely empty?
["Title: How Long Will It Take to Empty a Cylindrical Water Tank with a 0.5m/h Leak?", "When managing water storage systems like a cylindrical tank, understanding how fast water drains is crucial for maintenance, planning, and safety. Consider a large cylindrical tank with a radius of 3 meters and a height of 10 meters, completely filled with water. If a leak causes the water level to drop by 0.5 meters every hour, knowing the total time to fully drain the tank helps with emergency response and resource planning.", "### Understanding the Tank’s Volume", "To calculate how long it takes for the tank to empty, start by determining the total volume of water inside. The volume ( V ) of a cylinder is given by the formula:", "[\nV = \pi r^2 h\n]", "- Radius ( r = 3 ) meters\n- Height ( h = 10 ) meters", "Plugging in the values:", "[\nV = \pi \ imes (3)^2 \ imes 10 = \pi \ imes 9 \ imes 10 = 90\pi , \ ext{cubic meters}\n]", "This means the tank holds approximately ( 90\pi \approx 282.74 ) cubic meters of water initially.", "### Calculating the Volume Loss Per Hour", "The water level drops at 0.5 meters per hour. Since the tank’s cross-sectional area is constant, the volume lost each hour depends on how much vertical space drains. The surface area ( A ) of the tank’s top surface is:", "[\nA = \pi r^2 = \pi \ imes 9 = 9\pi , \ ext{square meters}\n]", "Each hour, with a drop of 0.5 meters, the volume drained per hour is:", "[\n\ ext{Volume loss per hour} = \ ext{Surface Area} \ imes \ ext{Height drop per hour} = 9\pi \ imes 0.5 = 4.5\pi , \ ext{cubic meters/hour}\n]", "### Determining Total Time to Empty", "To find the total time required to completely empty the tank, divide the total volume by the hourly loss:", "[\n\ ext{Time to empty} = \frac{90\pi}{4.5\pi} = \frac{90}{4.5} = 20 , \ ext{hours}\n]", "### Summary", "With a cylindrical tank of radius 3 meters and height 10 meters filled completely, a leak causing a 0.5-meter drop in water level per hour means the tank will be fully drained in exactly 20 hours. Understanding this timeline helps facility managers respond quickly, minimize water loss, and schedule repairs efficiently.", "---", "Key Takeaway:\nA 3-meter diameter cylindrical tank with a 10-meter height losing water at 0.5 meters per hour empties in 20 hours due to the consistent volume loss tied to its cross-sectional area. Use this calculation to plan for water maintenance and emergency monitoring."]
