A chemical company needs a new open-top cylindrical tank to store a chemical solution. The tank must have a volume of 50 m 3 50~\text{m}^3 . To minimize the material cost, the surface area of the tank (including the base but not the top) should be minimized. What should be the radius and height of the tank in meters?

Radius: 50 π 3 m \sqrt[3]{\frac{50}{\pi}}~\text{m} 3 π 50 <path d="M983 90 l0 -0 c4,-6.7,10,-10,18,-10 H400000v40 H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7 s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744 c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30 c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722 c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5 c53.7,-170.3,84.5,-266.8,92.5,-289.5z M1001 80h400000v40h-400000z"> m Height: 50 π 3 m \sqrt[3]{\frac{50}{\pi}}~\text{m} 3 π 50 <path d="M983 90 l0 -0 c4,-6.7,10,-10,18,-10 H400000v40 H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7 s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744 c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30 c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722 c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5 c53.7,-170.3,84.5,-266.8,92.5,-289.5z M1001 80h400000v40h-400000z"> m

Radius: 50 π 3 m \sqrt[3]{\frac{50}{\pi}}~\text{m} 3 π 50 <path d="M983 90 l0 -0 c4,-6.7,10,-10,18,-10 H400000v40 H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7 s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744 c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30 c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722 c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5 c53.7,-170.3,84.5,-266.8,92.5,-289.5z M1001 80h400000v40h-400000z"> m Height: 2 50 π 3 m 2\sqrt[3]{\frac{50}{\pi}}~\text{m}

Radius: 25 π 3 m \sqrt[3]{25\pi}~\text{m} Height: 25 π 3 m \sqrt[3]{25\pi}~\text{m}

Radius: 25 π 3 m \sqrt[3]{25\pi}~\text{m} Height: 50 π 3 m \sqrt[3]{\frac{50}{\pi}}~\text{m}

A chemical company needs a new open-top cylindrical tank to store...

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