#### 4Question: An industrial engineer designs a storage tank with a cylindrical section of height $ 3r $ and radius $ r $, and a hemispherical top of radius $ r $. What is the ratio of the hemispheres volume to the total volume of the tank?
["#### #### An industrial engineer designs a storage tank with a cylindrical section of height $ 3r $ and radius $ r $, and a hemispherical top of radius $ r $. What is the ratio of the hemispheres volume to the total volume of the tank? \nAs industrial efficiency gains momentum in U.S. engineering, innovative tank designs are drawing attention for combining form, function, and resource-conscious materials. This storage tank configuration—featuring a vertical cylindrical chamber rising three times the radius above a seamless hemispherical cap—reflects practical solutions to space optimization and material efficiency. The fusion of cylindrical capacity with hemispherical headquestions how volume is distributed across shapes, a topic increasingly relevant in manufacturing, logistics, and infrastructure planning. With growing focus on sustainable design, understanding the relative volume contribution of each section helps engineers and users alike balance storage needs with cost, weight, and structural integrity.", "Why #### An industrial engineer designs a storage tank with a cylindrical section of height $ 3r $ and radius $ r $, and a hemispherical top of radius $ r $? This combination is gaining traction due to its functional advantages. The cylindrical section supports bulk liquid or material containment, while the hemispherical top adds volume without excessive surface area, reducing construction material and enhancing pressure distribution. In sectors like chemical processing, mining, and renewable energy storage, such designs reduce footprint and improve load handling. This balance of form and function has sparked interest among engineers seeking optimized, space-efficient systems—especially in constrained urban or rugged terrains where maximum utility meets minimum space.", "How #### An industrial engineer designs a storage tank with a cylindrical section of height $ 3r $ and radius $ r $, and a hemispherical top of radius $ r $? The total stored volume equals the sum of the cylinder and hemisphere volumes. The cylindrical portion contributes $ V_{\ ext{cyl}} = \pi r^2 \cdot (3r) = 3\pi r^3 $, while the hemispherical cap holds $ V_{\ ext{hem}} = \frac{2}{3}\pi r^3 $. Adding them: \nTotal volume $ V_{\ ext{total}} = 3\pi r^3 + \frac{2}{3}\pi r^3 = \frac{11}{3}\pi r^3 $.", "Since the hemispherical section contributes $ \frac{2}{3}\pi r^3"]
