Ein zylindrischer Tank mit einem Radius von 3 Metern und einer Höhe von 10 Metern ist mit Wasser gefüllt. Wenn der Tank gekippt wird, bis das Wasser gerade die Basis bedeckt, wie viel Wasser befindet sich im Tank?
["Title: How Much Water Remains in a 3-Meter-Radius Cylindrical Tank When Tipped to Cover the Base?", "When managing water storage systems, one common question arises: what happens to the volume of water when a cylindrical tank is tilted so the water just reaches the base edges? In this article, we explore a precise calculation involving a cylindrical tank with a radius of 3 meters and a height of 10 meters, fully filled with water, then tilted until the water surface aligns exactly with the tank’s circular base.", "---", "### The Scenario", "- Tank: Cylinder\n- Radius ( r = 3 ) meters\n- Initial full height ( h = 10 ) meters\n- When the tank is upright, it holds a volume of water equal to its total capacity:\n [\n V_{\ ext{full}} = \pi r^2 h = \pi \ imes 3^2 \ imes 10 = 90\pi \ ext{ cubic meters}\n ]", "Now, imagine tilting the tank until the water just reaches the outer edge of the circular base on one side — forming a surface that touches both the top rim and the base edge. At this critical tilt, the water level forms a chord across the circular cross-section, and the geometry becomes key to computing the remaining volume.", "---", "### Geometry of Water in a Tilted Cylinder", "When the cylinder is tilted so the water touches the base edge at one rim and the top rim directly opposite, the water level becomes a horizontal chord cutting the circular cross-section. This split divides the circular area into two segments — and since the water just reaches the base edge, the fill is exactly half the area of the circular base in height projection — but volume depends on the three-dimensional shape formed.", "For a cylinder tilted to this point, a well-known geometric result applies: the volume of liquid in a vertically oriented cylinder, when tilted so the water surface passes through a diameter of the base, equals half the total volume.", "This arises because the water surface becomes a diameter of the circular base, and the filled region forms a half-cylinder in terms of volume, even though the shape remains curved.", "---", "### Why Volume is Half: A Derivation Insight", "Consider a vertical circular cross-section of radius ( r = 3 ) meters. When tilted so the water surface runs along a diameter, the filled region resembles half a circular segment extended through height. However, due to symmetry and the exact edge condition, the volume below that horizontal line (from base up to the top-selling point) integrates to exactly half the total cylindrical volume.", "Alternatively, think of slicing the cylinder horizontally — the surface at tilt forms one edge, and integrating the volume along height confirms that the area of the segment times length shows symmetry. The cumulative liquid volume is:", "[\nV_{\ ext{water}} = \frac{1}{2} \ imes \pi r^2 h = \frac{1}{2} \ imes \pi \ imes 9 \ imes 10 = 45\pi \ ext{ cubic meters}\n]", "---", "### Final Answer", "When the cylindrical tank with a 3-meter radius and 10-meter height is filled with water and then tilted until the water exactly covers the base from edge to edge, the remaining volume of water is:", "[\n\boxed{45\pi \ ext{ cubic meters}} \quad (\approx 141.37, \ ext{m}^3)\n]", "---", "### Practical Implications", "Understanding how volume transforms under tilt is essential for engineering applications such as mobile water tanks, storage tanks in uneven terrain, or safety assessments during spillage risk. This calculation helps design operators manage capacity accurately even after physical perturbations.", "---", "Keywords: cylindrical tank volume, tilted cylinder water amount, water in cylinder radius 3m, tank geometry, half-full cylinder volume, how much water tilts tank, water level calculation, fluid dynamics cylinder", "---", "Now you know — when geometry meets physics, even water finds its balanced form."]
