A standard soda can is roughly cylindrical and holds 355cm3 of liquid. What dimensions should the cylinder be to minimize the material needed to produce the can? Based on your dimensions, determine whether or not the standard can is produced to minimize the material costs.

curve(  2*3.14*x^2 + 710/x, from=1, to=10, n=300, xlab="radius", ylab="surface area", 
             col="blue", lwd=2, main="Surface area with respect to radius length"  )

Based on the graph, there is a minimum between 2 and 4

The volume of a cylindrical can is given by:

\(V= \pi r^2 h\)

Height of the can in terms of radius with \(355 cm^3\) as a restraint.

\(355 = \pi r^2 h\)

\(h = \frac{355}{\pi r^2}\)

Surface area of the can

\(SA_{can}=2\pi r^2 + 2\pi r h\)

In order to minimize, I’ll take the derivative and set to zero to find critical value.

\(\frac{dSA}{dr}=4\pi r -\frac{710}{r^2}\)

$ 4r= $

r = 3.84cm therefore h = 7.67

The US standard can is 4.83 in or 12.3 cm high, 2.13 in or 5.41 cm in diameter at the lid so the optimal dimensions are not used.