Optimal soda can a. Classical problem Find the radius and height of a cylindrical soda can with a volume of 354 cm3 that minimize the surface area. b. Real problem Compare your answer in part (a) to a real soda can, which has a volume of 354 cm3, a radius of 3.1 cm, and a height of 12.0 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 areas of the top and bottom are now twice their values in part (a)). Are these dimensions closer to the dimensions of a real soda can?
Solution 39E Step 1: (a) Consider a cylindrical soda can with the volume of cm . Let the radius of the cylinder be and eight be . Consider that the volume of the cylinder with radius and height is given by the following formula V = r h2 …(1) So, in this case volume is, V = r h = 354 h = r2 Now, the total surface area of the cylinder with radiusr and heighth is given by the following formula A = 2r +2rh ….(2) Step 2 The objective is to minimize the surface area. For minimizing the surface area first find teh critical point by differentiating A with respect to r and equate to 0 Differentiate the following equation with respect to For minimizing the surface area, equate to zero Put this value of in the following equation to find the value ofh Hence, the radius and height of the cylindrical soda can to minimize the surface area are r = 3.83cm and h = 7.67cm