A cylindrical tank with a radius of 4 meters and a height of 10 meters is filled with water. If a spherical ball with a radius of 2 meters is submerged in the tank, by how much will the water level rise?
["### How Much Will Water Level Rise in a Cylindrical Tank When a Spherical Ball Is Submerged?", "Understanding how submerged objects affect water levels in cylindrical containers is essential in fluid dynamics, hydrology, and engineering. In this article, we'll explore how a spherical ball with a 2-meter radius, when fully submerged in a cylindrical tank with a 4-meter radius and 10-meter height filled with water, influences the water level. Specifically, we’ll calculate by how much the water level rises.", "---", "### Tank and Ball Dimensions", "- Cylindrical tank radius: 4 meters\n- Tank height: 10 meters (not directly needed for level rise calculation, but provides context)\n- Submerged spherical ball radius: 2 meters", "The key focus is on the volume of water displaced by the sphere and how that displacement affects the water level in the wider tank.", "---", "### Calculating the Volume of the Submerged Ball", "The volume ( V ) of a sphere is given by the formula:", "[\nV = \frac{4}{3} \pi r^3\n]", "Substituting the ball’s radius ( r = 2 ) meters:", "[\nV_{\ ext{ball}} = \frac{4}{3} \pi (2)^3 = \frac{4}{3} \pi \cdot 8 = \frac{32}{3} \pi \ ext{ cubic meters}\n]", "---", "### Estimating Water Level Rise in the Cylindrical Tank", "The water level rise in the cylindrical tank is determined by dividing the displaced volume by the base area of the tank.", "The base area ( A ) of the tank is:", "[\nA = \pi R^2 = \pi (4)^2 = 16\pi \ ext{ square meters}\n]", "The rise in water level ( h ) is:", "[\nh = \frac{V_{\ ext{ball}}}{A} = \frac{\frac{32}{3} \pi}{16 \pi} = \frac{32}{3} \div 16 = \frac{32}{3 \cdot 16} = \frac{2}{3} \ ext{ meters}\n]", "---", "### Final Result", "When a spherical ball with a 2-meter radius is fully submerged in a cylindrical tank with a 4-meter radius, the water level rises by (\frac{2}{3}) meters, or approximately 0.67 meters.", "---", "### Practical Implications", "- This calculation helps engineers and architects estimate space requirements when placing underwater objects in cylindrical storage or containment tanks.\n- Simple volumetric displacement principles allow accurate predictions without complex modeling, saving time and resources in design and safety assessments.", "---", "### Summary", "- Cylinder radius: 4 m\n- Sphere radius: 2 m\n- Tank height: 10 m (reference, not affecting level rise)\n- Volume of sphere: ( \frac{32}{3}\pi , \ ext{m}^3 )\n- Tank base area: ( 16\pi , \ ext{m}^2 )\n- Water level rise: ( \frac{2}{3} ) meters (~0.67 m)", "This straightforward physics example highlights how displacement principles directly impact measured levels, essential for many real-world applications from pool design to industrial tanks.", "---", "Keywords: water level rise, cylindrical tank, submerged sphere, cylinder volume calculation, fluid displacement, 물 level rise calculation, cylindrical volume, submerged object displacement.", "---", "Understanding volume displacement helps us predict and control water level changes — a fundamental concept in hydrodynamics and engineering design."]
