Find the radius and height of a cylindrical soda can with a volume of 256cm^3 that minimize the surface area.B: Compare your answer in part A to a real soda can, which has a volume of 256cm^3, a radius of 2.8 cm, and a height of 10.7 cm, to conclude that real soda cans do not seem to have an optimal design. Then use the fact that real soda cans have a double thickness in their top and bottom surfaces to find the radius and height that minimizes the surface area of a real can (the surface area of the top and bottom are now twice their values in part A.B: New radius=?New height=?

Answer:A) Radius: 3.44 cm.Height: 6.88 cm.B) Radius: 2.73 cm.Height: 10.92 cm.Step-by-step explanation:We have to solve a optimization problem with constraints. The surface area has to be minimized, restrained to a fixed volumen.a) We can express the volume of the soda can as:[tex]V=\pi r^2h=256[/tex]This is the constraint.The function we want to minimize is the surface, and it can be expressed as:[tex]S=2\pi rh+2\pi r^2[/tex]To solve this, we can express h in function of r: [tex]V=\pi r^2h=256\\\\h=\frac{256}{\pi r^2}[/tex]And replace it in the surface equation[tex]S=2\pi rh+2\pi r^2=2\pi r(\frac{256}{\pi r^2})+2\pi r^2=\frac{512}{r} +2\pi r^2[/tex]To optimize the function, we derive and equal to zero[tex]\frac{dS}{dr}=512*(-1)*r^{-2}+4\pi r=0\\\\\frac{-512}{r^2}+4\pi r=0\\\\r^3=\frac{512}{4\pi} \\\\r=\sqrt[3]{\frac{512}{4\pi} } =\sqrt[3]{40.74 }=3.44[/tex]The radius that minimizes the surface is r=3.44 cm.The height is then[tex]h=\frac{256}{\pi r^2}=\frac{256}{\pi (3.44)^2}=6.88[/tex]The height that minimizes the surface is h=6.88 cm.b) The new equation for the real surface is:[tex]S=2\pi rh+2*(2\pi r^2)=2\pi rh+4\pi r^2[/tex]We derive and equal to zero[tex]\frac{dS}{dr}=512*(-1)*r^{-2}+8\pi r=0\\\\\frac{-512}{r^2}+8\pi r=0\\\\r^3=\frac{512}{8\pi} \\\\r=\sqrt[3]{\frac{512}{8\pi}}=\sqrt[3]{20.37}=2.73[/tex]The radius that minimizes the real surface is r=2.73 cm.The height is then[tex]h=\frac{256}{\pi r^2}=\frac{256}{\pi (2.73)^2}=10.92[/tex]The height that minimizes the real surface is h=10.92 cm.