Can Design There are many factors to consider in food packaging, including marketing, durability, cost and materials. In this problem we minimize the cost of materials for a can. Find the height and radius that minimizes the surface area of a can whose volume is 1 liter = 1000 cm3 . Solution We start by drawing a sketch of the can. The formula for its volume is (area of base) · (height) = πr2 h. Its surface consists of three parts — top, bottom and sides. The surface areas of the top and bottom are πr2 . The surface area of the sides is found by multiplying the circumference of the bottom by the height of the can; it is 2πrh. We now have the two equations: V = 1 liter SA = πr2 h = 2πr2 + 2πrh. We wish to minimize the surface area, which suggests taking a derivative. Since h is dependent on r we cannot directly take the derivative of SA directly yet. We solve the volume equation for h: = πr2 h 1 h = . πr2 1 We can now plug this expression in to the surface area equation and take the derivative with respect to r. = 2πr2 + 2πrh = 2πr2 + 2πr SA = 2πr2 + d SA dr = SA 1 πr2 2 r 2 4πr − 2 r To ﬁnd the critical points, we set the derivative equal to zero and solve for r: d SA = dr 2 4πr − 2 = r 4πr = 0 0 2 r2 r3 = r = 1 2π 1 √ . 3 2π 1 We now know that there is a critical point at r = √ , but we do not know 3 2π if this corresponds to a minimum surface area. We must also consider the “end points” of the graph of our function. As r approaches 0, the can is increasingly tall and thin; the area of the sides of the can approaches inﬁnity. This is not the best solution. As r increases the surface area 2πr2 + 2r goes to inﬁnity, so the minimum is not at this endpoint either. 1 We conclude that the critical point r = √ probably represents the min 3 2π imum surface area. To be sure, we calculate h and estimate the size of the can. h = = h = 1 πr2 1 1 π( √ )2 3 2π � 3 4 π � 1 One liter is 1000 cubic centimeters, so r = 3 2π liters ≈ 5 cm and h ≈ 11 cm. This is about the same size as the bottom half of a 2 liter bottle, so seems like a reasonable answer.